UC-NRLF 


SB    ESS    755 


INTENTIONAL  GEOMETRY: 


A  SERIES  OF  PROBLEMS, 

INTENDED  TO 

FAMILIARIZE  THE  PUPIL  WITH  GEOMETRICAL  CONCEPTIONS^ 
AND  TO  EXERCISE  HIS  INVENTIVE  FACULTY. 


BY 

WILLIAM  GEORGE  SPENCER, 


WITH  A  PREFATORY   NOTE 

BY  HERBERT  SPENCER. 


NEW  YOFK  . :  -  CINCINNATI  - :  •  CHICAGO : 
AMERICAN     BOOK     COMPANY 


PRIMER    SERIES. 

SCIENCE   PRIflERS. 

HUXLEY'S  INTRODUCTORY  VOLUME. 

ROSCOE'S  CHEMISTRY. 

STEWART'S  PHYSICS. 

GEIKIE'S  GEOLOGY. 

LOCKYER'S   ASTRONOMY. 

HOOKER'S   BOTANY 

FOSTER    AND    TRACY'S    PHYSIOLOGY   AND 

HYGIENE. 

GEIKIE'S   PHYSICAL  GEOGRAPHY 
HUNTER'S   HISTORY  OF  PHILOSOPHY. 
LUPTON'S  SCIENTIFIC  AGRICULTURE. 
JEVONS'S   LOGIC. 

SPENCER'S  INVENTIONAL   GEOMETRY. 
JEVONS'S   POLITICAL  ECONOMY. 
TAYLOR'S   PIANOFORTE  PLAYING. 
PATTON'S    NATURAL    RESOURCES   OF  THE 

UNITED  STATES. 

HISTORY   PRIHERS. 

WENDEL'S  HISTORY  OF  EGYPT. 
FREEMAN'S  HISTORY  OF  EUROPE. 
FYFFE'S  HISTORY  OF  GREECE. 
CREIGHTON'S   HISTORY  OF  ROME. 
MAHAFFY'S  OLD  GREEK  LIFE. 
WILKINS'S    ROMAN    ANTIQUITIES. 
TIGHE'S  ROMAN   CONSTITUTION. 
ADAMS'S  MEDIAEVAL  CIVILIZATION. 
YONGE'S   HISTORY  OF   FRANCE. 
GROVE'S  GEOGRAPHY. 

LITERATURE  PRIflERS. 

BROOKE'S  ENGLISH  LITERATURE. 

WATKINS'S  AMERICAN   LITERATURE. 

DOWDEN'S   SHAKSPERE. 

ALDEN'S  STUDIES   IN  BRYANT. 

MORRIS'S   ENGLISH  GRAMMAR. 

MORRIS     AND    BOWEN'S    ENGLISH    GRAM- 

.,MAR    EXERCISES. 
NfCHQL'S  ENGXL^H  JCGty  POSITION. 
PEICEXS.  PHILOLOGY. 
JEBB'S  GREEK   LITERATURE. 
GLADSTONE'S   HOMER.      , 

$ « fJLAS'SIC AL  GEOGRAPHY. 


SPENCER— INV.  GEOM. 

COPYRIGHT,  1876,  BY  D.  ArPLETON  &  CO. 
w.  P.  13 


INTRODUCTION 


WHEN  it  is  considered  that  by  geometry  the 
architect  constructs  our  buildings,  the  civil  en- 
gineer our  railways ;  that  by  a  higher  kind  of 
geometry,  the  surveyor  makes  a  map  of  a 
county  or  of  a  kingdom ;  that  a  geometry  still 
higher  is  the  foundation  of  the  noble  science  of 
the  astronomer,  who  by  it  not  only  determines 
the  diameter  of  the  globe  he  lives  upon,  but  as 
well  the  sizes  of  the  sun,  moon,  and  planets, 
and  their  distances  from  us  and  from  each 
other ;  when  it  is  considered,  also,  that  by  this 
higher  kind  of  geometry,  with  the  assistance  of 
a  chart  and  a  mariner's  compass,  the  sailor  navi- 
gates the  ocean  with  success,  and  thus  brings  all 
nations  into  amicable  intercourse — it  will  surely 
be  allowed  that  its  elements  should  be  as  acces- 
sible as  possible. 


Geometry  may  be  divided  into  two  parts- 
practical  and  theoretical :  the  practical  bearing 
a  similar  relation  to  the  theoretical  that  arith- 
metic does  to  algebra.  And  just  as  arithmetic 
is  made  to  precede  algebra,  should  practical  ge- 
ometry be  made  to  precede  theoretical  geome- 
try. 

Arithmetic  is  not  undervalued  because  it  is 
inferior  to  algebra,  nor  ought  practical  geome- 
try to  be  despised  because  theoretical  geometry 
is  the  nobler  of  the  two. 

However  excellent  arithmetic  may  be  as  an 
instrument  for  strengthening  the  intellectual 
powers,  geometry  is  far  more  so ;  for  as  it  is 
easier  to  see  the  relation  of  surface  to  surface 
and  of  line  to  line,  than  of  one  number  to  an- 
other, so  it  is  easier  to  induce  a  habit  of  reason- 
ing by  means  of  geometry  than  it  is  by  means 
of  arithmetic.  If  taught  judiciously,  the  collat- 
eral advantages  of  practical  geometry  are  not 
inconsiderable.  Besides  introducing  to  our  no- 
tice, in  their  proper  order,  many  of  the  terms 
of  the  physical  sciences,  it  offers  the  most  favor- 
able means  of  comprehending  those  terms,  and 


INTRODUCTION.  7 

impressing  them  upon  the  memory.  It  educates 
the  hand  to  dexterity  and  neatness,  the  eye  to 
accuracy  of  perception,  and  the  judgment  to 
the  appreciation  of  beautiful  forms.  These  ad- 
vantages alone  claim  for  it  a  place  in  the  educa- 
tion of  all,  not  excepting  that  of  women.  Had 
practical  geometry  been  taught  as  arithmetic  is 
taught,  its  value  would  scarcely  have  required 
insisting  on.  But  the  didactic  method  hitherto 
used  in  teaching  it  does  not  exhibit  its  powers 
to  advantage. 

Any  true  geometrician  who  wjll  teach  practi- 
cal geometry  by  definitions  and  questions  there- 
on, will  find  that  he  can  thus  create  a  far  great- 
er interest  in  the  science  than  he  can  by  the 
usual  course ;  and,  on  adhering  to  the  plan,  he 
will  perceive  that  it  brings  into  earlier  activity 
that  highly-valuable  but  much-neglected  power, 
the  power  to  invent.  It  is  this  fact  that  has  in- 
duced the  author  to  choose  as  a  suitable  name 
for  it,  the  inventional  method  of  teaching  prac 
tical  geometry. 

He  has  diligently  watched  its  effects  on  both 
*exes,  and  his  experience  enables  him  to  say 


g  INTRODUCTION. 

that  its  tendency  is  to  lead  the  pupil  to  rely  on 
his  own  resources,  to  systematize  his  discoveries 
in  order  that  he  may  use  them,  and  to  gradually 
induce  such  a  degree  of  self-reliance  as  enables 
him  to  prosecute  his  subsequent  studies  with  sat- 
isfaction: especially  if  they  st'onld  happen  to 
be  such  studies  as  Euclid's  "Elen-ents,"  the  nse 
of  the  globes,  or  perspective. 

A  word  or  two  as  to  using  the  definitions 
and  questions.  Whether  they  relate  to  the 
mensuration  of  solids,  or  surfaces,  or  of  lines ; 
\vhether  they  Belong  to  common  square  meas- 
ure, or  to  duodecimals ;  or  whether  they  apper- 
tain to  the  canon  of  trigonometry ;  it  is  not  the 
author's  intention  that  the  definitions  should  be 
learned  by  rote ;  but  he  recommends  that  the 
pupil  should  give  an  appropriate  illustration  oi 
each  as  a  proof  that  he  understands  it. 

Again,  instead  of  dictating  to  the  pupil  how 
to  construct  a  geometrical  figure — say  a  square 
—and  letting  him  rest  satisfied  with  being  able 
to  construct  one  from  that  dictation,  the  author 
has  so  organized  these  questions  that  by  doing 
justice  to  each  in  its  turn,  the  pupil  finds  that, 


INTRODUCTION.  9 

when  he  comes  to  it,  he  can  construct  a  square 
without  aid. 

The  greater  part  of  the  questions  accompany- 
ing the  definitions  require  for  their  answers  ge- 
ometrical figures  and  diagrams,  accurately  con- 
structed by  means  of  a  pair  of  compasses,  a  scale 
of  equal  parts,  and  a  protractor,  while  others 
require  a  verbal  answer  merely.  In  order  to 
place  the  pupil  as  much  as  possible  in  the  state 
in  which  Nature  places  him,  some  questions 
have  been  asked  that  involve  an  impossibility. 

Whenever  a  departure  from  the  scientific 
order  of  the  questions  occurs,  such  departure 
has  been  preferred  for  the  sake  of  allowing 
time  for  the  pupil  to  solve  some  difficult  prob- 
lem ;  inasmuch  as  it  tends  far  more  to  the  for- 
mation of  a  self-reliant  character,  that  the  pupil 
should  be  allowed  time  to  solve  such  difficult 
problem,  than  that  he  should  be  either  hurried 
or  assisted. 

The  inventive  power  grows  best  in  the  sun 
shine  of  encouragement.  Its  first  shoots  are 
tender.  Upbraiding  a  pupil  with  his  want  of 
skill,  acts  like  a  frost  upon  them,  and  materially 


10  INTRODUCTION. 

checks  their  growth.  It  is  partly  on  account  of 
the  dormant  state  in  which  the  inventive  power 
is  found  in  most  persons,  and  partly  that  very 
young  beginners  may  not  feel  intimidated,  that 
the  introductory  questions  have  teen  made  so 
very  simple. 


TO  THE  PUPIL 


WHEN  it  is  found  desirable  to  save  time,  omit 
copying  the  definitions ;  but  when  time  can  be 
spared,  copy  them  into  the  trial-book,  to  im- 
press the  terms  on  the  memory. 

In  constructing  a  figure  that  you  know,  use 
arcs  if  you  prefer  them ;  but,  in  all  your  at- 
tempts to  solve  a  problem,  prefer  whole  circles 
to  arcs.  Circles  are  suggestive,  arcs  are  not. 

Always  have  a  reason  for  the  method  you 
adopt,  although  you  may  not  be  able  to  express 
it  satisfactorily  to  another.  Such,  for  example, 
as  this :  If  from  one  end  of  a  line,  as  a  centre,  I 
describe  a  circle  of  a  certain  size,  and  then  from 
the  other  end  of  the  line,  as  another  centre,  I 
describe  another  circle  of  the  same  size,  the 
points  where  those  circles  intersect  each  other, 
if  they  intersect  at  all,  must  have  the  same  rela 


12  TO  THE  PUPIL 

tion  to  one  end  of  such  line  which  they  have  to 
the  other. 

The  most  improving  method  of  entering  the 
solutions  is  to  show,  in  a  first  figure,  all  the  cir- 
cles in  full  by  which  you  have  arrived  at  the 
solution,  and  to  draw  a  second  figure  in  ink, 
without  the  circles. 

It  is  not  so  much  the  problems  which  you 
are  assisted  in  performing,  as  the  problems  you 
perform  yourself,  that  will  improve  your  talents 
and  benefit  your  character.  Refrain,  then,  from 
looking  at  the  constructions  invented  by  other 
persons — at  least  till  you  have  discovered  a 
construction  of  your  own.  The  less  assistance 
you  seek  the  less  you  will  require,  and  the  less 
you  will  desire. 

As  the  power  to  invent  is  ever  varying  in 
the  same  person,  and  as  no  two  persons  have 
that  power  equally,  it  is  better  not  to  be  anxious 
about  keeping  pace  with  others.  Indeed,  all 
your  efforts  should  be  free  from  anxiety.  Pleas- 
urable efforts  are  the  most  effective.  Be  as- 
sured that  no  effort  is  lost,  though  at  the  time 
it  may  appear  so  You  may  improve  more 


TO  THE  PUPIL.  13 

while  studying  one  problem  that  is  rather  intri- 
cate to  you,  than  while  performing  several  that 
are  easy.  Dwell  upon  what  the  immortal  New- 
ton said  of  his  own  habit  of  study.  "  I  keep," 
says  he,  "  the  subject  constantly  before  me,  and 
wait  till  the  first  dawnings  open  by  little  and 
little  into  a  full  and  clear  light." 


INVENTIONAL   GEOMETRY. 


THE  science  of  relative  quantity,  solid,  su- 
perficial, and  linear,  is  called  Geometry,  and 
the  practical  application  of  it,  Mensuration. 
Thus  we  have  mensuration  of  solids,  mensura- 
tion of  surfaces,  and  mensuration  of  lines  ;  and 
to  ascertain  these  quantities  it  is  requisite  that 
we  should  have  dimensions. 

The  top,  bottom,  and  sides  of  a  solid  body, 
as  a  cube,1  are  called  its  faces  or  surfaces,1  and 
the  edges  of  these  surfaces  are  called  lines. 

The  distance  between  the  top  and  bottom 
of  the  cube  is  a  dimension  called  the  height, 
depth,  or  thickness  of  the  cube ;  the  distance 
between  the  left  face  and  the  right  face  is  anoth- 

1  The  most  convenient  form  for  illustration  is  that  of  the 
cubic  inch,  which  is  a  solid,  having  equal  rectangular  surfaces 
*  A  surface  is  sometimes  called  a  superficies. 


16  INVENTIONAL   GEOMETRY. 

er  dimension,  called  the  breadth  or  width  ;  and 
the  distance  between  the  front  face  and  the 
back  face  is  the  third  dimension,  called  the 
length  of  the  cube. 

Thus  a  cube  is  called  a  magnitude  of  three 
dimensions. 

The  three  terms  most  commonly  applied  to 
the  dimensions  of  a  cube  are  length,  breadth, 
and  thickness. 

1.  Place  a  cube  with  one  face  flat  on  a  table, 
and  with  another  face  toward  you,  and  say  which 
dimension   you   consider  to  be  the  thickness, 
which  the  breadth,  and  which  the  length. 

2.  Show  to  what  objects  the  word  height  is 
more  appropriate,  and  to  what  objects  the  word 
depth,  and  to  what  the  word  thickness. 

As  a  surface  has  nc  thickness,  it  has  two  di- 
mensions only,  length  and  breadth.  Thus  a 
surface  is  called  a  magnitude  of  two  dimen- 
sions. 

3.  Show  how  many  faces  a  cube  has.1 

1  The  surfaces  of  a  cube  are  considered  to  be  plane  tup 
fecet. 


1NVE8TWNAL   GEOMETRY.  17 

When  a  surface  is  such,  that  a  line  placed 
anywhere  upon  it  will  rest  wholly  on  that  sur-; 
face,  such  surface  is  said  to  he  a  plane  sur- 
face.1 

As  a  line  has  neither  breadth  nor  thickness, 
it  has  one  dimension  only,  that  of  length. 

Thus  a  line  is  called  a  magnitude  of  one 
dimension. 

4.  Count  how  many  lines  are  formed  on  a 
cube  by  the  intersection  of  its  six  plane  surfaces. 

If  that  which  has  neither  breadth,  nor  thick 
ness,  but  length  only,  can  be  said  to  have  any 
form,  then  a  line  is  such,  that  if  it  were  turned 
upon  its  extremities,  each  part  of  it  would  keep 
its  own  place  in  space. 

We  cannot  with  a  pencil  make  a  line  on  pa- 
per— we  represent  a  line. 

The  boundaries  or  ends  of  a  line  are  called 
points,  and  the  intersection  of  two  lines  gives 
a  point. 

As  a  point  has  neither  length,  breadth,  no? 

1  When  the  word  line  is  used  in  these  definitions  and  que* 
lions  a  straight  line  is  always  meant 


IS  INVENTION AL   GEOMETRY. 

thickness,  it  is  said  to  have  no  dimension.     It 
has  position  only. 

A  point  is  therefore  not  a  magnitude. 

5.  Name  the  number  of  points  that  are  made 
by  the  intersection  of  the  twelve  lines  of  a  cube 

We  cannot  with  a  pencil  make  a  point  on 
paper — we  represent  a  point. 

When  any  two  straight  lines  meet  together 
from  any  other  two  directions  than  those  which 
are  perfectly  opposite,  they  are  said  to  make  an 
angle. 

And  the  point  where  they  meet  is  called  the 
angular  point. 

Thus  two  lines  that  meet  each  other  on  a 
cube  make  an  angle. 

6.  Represent  on  paper  a  rectilineal  angle. 

7.  Can  two  lines  meet  together  without  be- 
ing in  the  same  plane  I 

8.  Point  out  two  lines  on  a  cube  that  exist 
on  the  same  surface,  and  yet  do  not  make  an 
angle, 

9.  Name  the  number  of  plane  angles  on 


INVENTWXAL   GEOMETRY.  19 

the  six  surfaces  of  a  cube,  and  the  number  of 
angular  points,  and  say  why  the  angular  points 
are  fewer  than  the  plane  angles. 

The  meeting  of  two  plane  surfaces  in  a  line 
--for  example,  the  meeting  of  the  wall  of  a 
room  with  the  floor,  or  the  meeting  of  two  of 
the  surfaces  of  a  cube — is  called  a  dihedral 
angle.1 

10.  Say  how  many  dihedral  angles  a  cube 
has. 

The  corner  made  by  the  meeting  of  three 
or  more  plane  surfaces  is  called  a  solid  angle. 

11.  Say  how  many  solid  angles  there  are  in 
a  cube. 

When  a  surface  is  such  that  a  line,  when 
resting  upon  it  in  any  direction,  will  be  touched 
by  it  toward  the  middle  of  the  line  only,  and 
not  at  both  ends,  such  surface  is  called  a  convex 
surface. 

12.  Give  an  example  of  a  convex  surface. 
When  a  surface  is  such  that  a  line  while  rest- 
ing upon  it,  in  any  direction,  will  be  touched  by 

1  Dihedral  means  two-surfaced. 


20  INVENTION AL   GEOMETRY. 

it  at  the  ends,  and  not  toward  the  middle  of  the 
line,  such  surface  is  called  a  concave  surface. 

13.  Give  an  example  of  a  concave  surface. 
A  simple  curve  is  such,  that  on  being  turned 

on  its  extremities,  every  point  along  it  will 
change  its  place  in  space;  so  that,  in  a  simple 
curve,  no  three  points  are  in  a  straight  line. 

14.  Give  an  example  of  a  simple  curve. 
Lines  or  curves  grouped  together  by  way  of 

illustration,  or  for  ornament,  without  regard  to 
magnitude  or  surface,  take  the  name  of  dia- 
grams. 

15.  Give  an  example  of  a  diagram. 
When  a  surface  l  is  spoken  of  with  regard  to 

its  form  and  size,  it  takes  the  name  of  figure. 

If  the  boundaries  of  a  surface  are  straight 
lines,  the  figure  is  called  a  rectilinear  figure,  and 
each  boundary  is  called  a  side. 

Thus  we  have  rectilinear  figures  of  four 
aides,  of  five  sides,  of  six  sides,  etc. 

16.  Make  a  few  rectilinear  figures. 

1  In  the  definitions  and  questions  of  this  work,  when  »hi 
trord  surface  is  used,  a  plane  surface  is  meant 


GEOMETRY.    .  %\ 

When  a  surface  is  inclosed  by  one  curve,  it 
is  called  a  curvilinear  figure,  and  the  boundary 
is  called  its  circumference. 

17.  Make  a  curvilinear  figure  with  one  curve 
for  its  boundary,  and  in  it  write  its  name,  and 
around  it  the  name  of  its  boundary. 

18.  Make  a  curvilinear  figure  with  more 
than  one  curve  for  its  boundaries. 

A  figure  bounded  by  a  line  and  a  curve,  or 
by  more  lines  and  more  curves  than  one,  ifl 
called  a  mixed  figure. 

19.  Make  a  mixed   figure,  having  for  its 
boundaries  a  line  and  a  curve. 

20.  Make  a  mixed   figure,  having  for  its 
boundaries  one  line  and  two  curves. 

21.  Make  a  mixed  figure,  having  for  its 
noundaries  one  curve  and  two  lines. 

When  a  figure  has  a  boundary  of  such  a 
form  that  all  lines  drawn  from  a  certain  point 
within  it  to  that  boundary  are  equal  to  one 
another,  such  figure  is  called  a  circle,  and  such 
point  is  called  the  centre  of  that  circle ;  and 


32  IJSVENT10NAL 

the  boundary  is  called  the  circumference  of  the 
circle,  and  the  equal  lines  drawn  from  the  cen- 
tre to  the  circumference  are  called  the  radii  of 
the  circle. 

22.  Make  four  circles.     On  the  first  write 
its  name.     Around  the  outside  of  the  second, 
write  the  name  of  the  boundary.     In  the  third, 
write  against  the  centre  its  name.     And  be- 
tween the  centre  and  the  circumference  of  the 
fourth  circle,  draw  a  few  radii  and  write  on  each 
its  name. 

23.  Can  you  place  two  circles  to  touch  each 
other  at  2  particular  point  ? 

24:.  Can  you  place  three  circles  in  a  row, 
and  let  each  circle  touch  the  one  next  to  it  ? 

A  part  of  the  circumference  of  a  circle  is 
called  an  arc. 

When  the  circumference  of  a  circle  is  di 
vided  into  two  equal  arcs,  each  arc  is  called  a 
semi-circumference. 

All  arcs  of  circles  which  extend  beyond  a 
aemi-circumference  are  called  greater  arcs. 


INVENT10KAL    GEOMETRY.  23 

All  arcs  of  circles  that  are  not  so  great  as  a 
semi-circumference  are  called  less  arcs. 

A  line  that  joins  the  extremities  of  an  arc  is 
called  the  chord  of  that  arc. 

When  two  radii  connect  together  any  two 
points  in  the  circumference  of  a  circle  which  are 
on  exactly  the  opposite  sides  of  the  centre,  they 
make  a  chord,  which  is  called  the  diameter  of 
the  circle,  and  such  diameter  divides  the  circle 
into  two  equal  segments,1  which  take  the  name 
of  semicircles. 

25.  Make  a  circle,  and  in  it  draw  two  radii 
in  such  a  position  as  to  divide  it  into  two  equal 
parts,  and  write  on  each  part  its  specific  name. 

All  segments  of  a  circle  which  occupy  more 
lhan  a  semi-circle  are  called  greater  segments. 

26.  Make  a  greater  segment,  and  on  it  write 
its  name. 

27.  Make  a  greater  segment,  and  on  the  out- 
side of  each  of  its  boundaries  write  its  name, 

:  The  word  segment  means  a  piece  cut  off:  thus  we  have 
•egments  of  a  line  and  segments  of  a  sphere,  as  wej]  ae  seg- 
ments of  a  circle. 


*4  INVENTIONAL   QEOMBT&*. 

All  segments  of  a  circle  that  do  not  contain 
so  much  as  a  semi-circle  are  callevl  less  segmenta 

28.  Make  a  less  segment,  and  in  it  write  its 
name. 

29.  Make  a  less  segment,  and  on  the  outside 
of  each  of  its  boundaries  write  its  name. 

30.  Can  you  cut  from  a  circle  more  than  one 
greater  segment  ? 

31.  Can  you  cut  from  a  circle  more  than  one 
less  segment  J 

32.  Place  two  circles  so  that  the  circumfer- 
ence of  each  may  rest  upon  the  centre  of  the 
other,  and  show  that  the  curved  figure  common 
to  both  circles  consists  of  two  segments,  and 
may  be  called  a  double  segment. 

33.  In  how  many  ways  can  you  divide  a 
double  segment  into  two  equal   and  similar 
parts  ? 

34.  In  how  many  ways  can  you  divide  a 
double   segment  into  four  equal   and  similar 
parts  ? 

35.  Can  you  make  two  angles  with  two  lines  f 


INVENTIONAL   GEOMETRY.  25 

When  two  lines  are  so  placed  as  to  make  two 
angles,  one  of  the  lines  is  said  to  stand  upon 
the  other,  and  the  angles  they  thus  make  are 
called  adjacent  angles. 

36.  Make  two  unequal  adjacent  angles  with 
two  lines. 

When  one  line  stands  upon  another  line,  in 
such  a  direction  as  to  make  the  adjacent  angles 
equal  to  one  another,  then  each  of  these  angles 
is  called  a  right  angle. 

37.  Make  two  equal  adjacent  angles,  and  in 
each  angle  write  its  proper  name. 

Either  of  the  sides  of  a  right  angle  is  said  to 
be  perpendicular  to  the  other ;  and  the  one  to 
which  the  other  is  said  to  be  perpendicular  is 
called  the  base. 

38.  Make  a  right  angle,  and  against  the  sides 
of  the  right  angle  write  their  respective  names. 

-?  39.  Can   you   make   three   angles   with   two 
lines  ? 

y  40.  Can  you  make  four  angles  with  two  lines  ? 

-x  41.  Can   you   make   more   than  four   angles 
with  two  lines  ?         3 


26  INVJSNTIOJfAL   &JSOM£TJRT. 

42.  Can  you  divide  a  line  into  two  equal 
parts? 

43.  Can  you  divide  an  arc  into  two  eqnaJ 
parts? 

You  have  been  told  that  figures  bounded  by 
lines  are  called  linear  figures. 

44.  Make  a  linear  figure  having  the  fewest 
boundaries  possible,  and  in  it  write  its  name, 
and  say  why  such  figure  claims  that  name.1 

When  a  figure  has  for  its  boundaries  three 
equal  lines,  it  is  called  an  equilateral  triangle.* 

45.  Can  you  make  an  equilateral  triangle? 

46.  Can  you   with  three  lines  make  two 
angles,  three,  four,  five,  six,  seven,  eight,  nine, 
ten,  eleven,  twelve,  thirteen  ? 

47.  Can  you  so  place  two  equilateral  trian- 
gles that  one  side  of  one  of  them  may  coincide 
with  one  side  of  the  other  ? 

48.  Can  you  divide  an  equilateral  triangle 
into  two  parts  that  shall  be  equal  to  each  othei 
and  similar  to  each  other? 

1  Triangles  are  also  called  trilateral^ 

"  Equilateral  triangles  are  also  called  tngrau. 


INVENT10NAL  GEOMETRY.       -  37 

>  49.  Can  you  draw  one  line  perpendicular  to 
another  line,  from  a  point  that  is  in  the  line  but 
not  in  the  middle  of  it  ? 

The  figure  formed  by  two  radii  and  an  arc  is 
jailed  a  sector. 

When  a  circle  is  divided  into  four  equal  sec- 
tors, each  of  such  sectors  takes  the  name  of  quad- 
rant. 

-?  50.  Divide  a  circle  into  four  equal  sectors, 
and  write  upon  each  sector  its  specific  name. 

51.  Make  a  set  of  quadrants,  and  write  in 
each  angle  its  specific  name. 

To  compare  sectors  of  different  magnitudes 
with  each  other,  geometricians  have  found  it 
3onvenient  to  imagine  every  circle  to  be  divided 
into  three  hundred  and  sixty  equal  sectors ;  and 
a  sector  consisting  of  the  three  hundred  and  six- 
tieth part  of  a  circle,  they  have  called  a  degree. 
An  arc,  therefore,  of  such  a  sector  is  an  arc  of  a 
degree ; '  and  the  angle  of  such  a  sector  is  an 
angle  of  a  degree. 

1  A  degree  of  a  circle  is  concisely  marked  thus  (1°)  Thir 
ty  degree*  thus  (30°).  Thirty-five  degrees  thus  (85°). 


28  INVENTION AL*  GEOMETRY. 

52.  Make  a  set  of  quadrants,  and  write  in 
each  angle  bow  many  degrees  it  contains. 

All  angles  greater  or  less  than  the  angle  oi 
a  quadrant  are  called  oblique  angles. 

When  an  oblique  angle  is  less  than  a  quad 
rantal  angle,  that  is  less  than  a  right  angle,  that 
is  less  than  an  angle  of  90°,  it  is  called  an  acute 
angle. 

53.  Make  an  acute  angle. 

When  an  oblique  angle  has  more  degrees  in 
it  than  90°,  and  less  than  180°,  it  is  called  an 
obtuse  angle. 

54.  Make  an  obtuse  angle. 

55.  Make  an  acute-angled  sector. 

56.  Make  an  obtuse-angled  sector. 

When  a  sector  has  an  arc  of  180°,  the  radii 
tormdng  with  each  other  one  straight  line,  it  has 
the  same  claim  to  be  called  a  sector  as  it  has  to 
be  called  a  segment,  and  yet  it  seldom  takes  the 
name  of  either,  being  generally  called  a  semi- 
circle. 

57.  Make  three  sectors,  each  containing  180°, 


INVENTION  A  L   GEOMETRY.         .  29 

nud  write  in  each  sector  a  different  name,  and 
yet  an  appropriate  one. 

A  sector  which  has  an  arc  greater  than  a 
semi-circumference  is  said  to  have  a  reentrant 
angle. 

58.  Make  a  reentrant-angled  sector. 

59.  Say  to  which  class  of  sectors  the  degree 
belongs. 

You  have  halved  a  line,  and  you  have  halved 
an  arc. 

60.  Can  you  divide  a  segment  into  two  parts 
that  shall  be  equal  to  each  other,  and  similar  to 
each  other  ? 

61.  Can  you  divide  a  sector  into  two  parts 
that  shall  be  equal  to  each  other,  and  similar  to 
each  other  ? 

It  is  said  by  some,  the  circumference  of  a 
circle  is  3  times  its  own  diameter ;  by  others, 
more  accurate,  that  it  is  3^  times  its  own  diain 
eter. 

62.  Say  how  you  would  determine  the  ratio 
the  circumference  of  a  circle  bears  to  its  diara 


30  *NVENTIONAL   GEOMETRY. 

eter,  and  say  also  what  you  make  the  ratio  to 
be. 

You  have  divided  a  line,  an  arc,  a  segment, 
and  a  sector,  into  two  equal  parts. 

63.  Can  you  divide  an  angle  into  two  equal 


When  a  triangle  has  two  only  of  its  sides  of 
equal  length  it  is  called  an  isosceles  triangle, 

64.  Make  an  isosceles  triangle. 

When  a  triangle  has  all  its  sides  of  different 
lengths  it  takes  the  name  of  scalene. 

65.  Make  a  scalene  triangle. 

When  a  triangle  has  one  of  its  angles  a  right 
angle,  it  is  called  a  right-angled  triangle. 

66.  Make  a  right-angled  triangle. 

When  a  triangle  has  each  of  its  angles  less 
than  a  right  angle,  and  all  different  in  size,  it  is 
called  a  common  acute-angled  triangle. 

67.  Make  a  common  acute-angled  triangle. 
When  a  triangle  has  one  of  its  angles  obtuse, 

it  is  called  an  obtuse-angled  triangle. 

68.  Make  an  obtuse-angled  triangle 


INVENTIONAL    VE'OMETRY.         .  31 

In  describing  the  properties  of  a  triangle  it 
is  not  unusual  to  mark  each  angular  point  of  the 
triangle  with  a  letter. 

Thus  the  accompanying  triangle  is  called  the 
triangle  A  B  C,  and  C 

the   sides  are  called 
A  B,  B  C,  and  A  C, 
and  the  three  angles 
are  called  A,  B,  C,  or  the  angles  C  A  B,  A  B  C, 
A  CB. 

69.  Can  you  make  an  isosceles  triangle  with- 
out using  more  than  one  circle  ? 

When  two  lines  do  not  meet  either  way, 
though  produced  ever  so  far,  they  are  said  to  be 
parallel.1 

70.  Draw  two  parallel  lines. 

71.  Can    you    draw   one    line    parallel    to 
another,  and  let  the  two  be  an  inch  apart  ? 

72.  Can  you  place  two  equal  sectors  so  that 
one  corresponding  radius  of  each  sector  may  be 
in  one- line,  and  so  that  their  angles  may  point 
the  same  way  ? 

1  Of  course  it  means  two  lines  in  the  same  plane. 


£%  INTENTIONAL   GEOMETRY. 

73.  Upon  the  same  side  of  the    same  line, 
piace  two  angles  that  shall  be  equal  to  each 
other,  and  let  each  angle  face  the  same  way. 

When  two  circles  have  the  same  centre,  they 
are  called  concentric  circles. 

74.  Make  three  concentric  circles. 

When  two  circles  have  not  the  same  centre, 
and  one  of  them  is  within  the  other,  they  are 
called  eccentric  circles. 

75.  Make  two  eccentric  circles. 

76.  Draw  one  line  parallel  to  another  line, 
and  let  it  pass  through  a  given  point. 

All  figures  that  have  four  sides  take  the 
name  of  quadrilaterals.1 

Of  quadrilaterals  there  are  six  varieties,  con- 
sisting of  quadrilaterals  that  have  their  oppo- 
site sides  parallel,  which  are  called  parallelo- 
grams; and  quadrilaterals  that  have  not  their 
sides  parallel,  which  are  called  trapeziums. 

Of  parallelograms  there  are  four  kinds : 
parallelograms,  which  have  all  the  sides  equal, 
and  all  the  angles  equal,  called  squares.  Paral 

1  Figures  of  four  sides  are  also  called  quadrangles. 


1NVEXT1ONAL   GEOMETRY.  33 

iclograms  which  have  the  sides  equal,  but  the 
angles  not  all  equal,  called  rhombuses.  Parallel- 
ograms which  have  all  their  angles  equal,  but 
their  sides  not  all  equal,  called  rectangles ;  and 
parallelograms  which  have  neither  the  sides  all 
equal,  nor  the  angles  all  equal,  called  rhom- 
boids. 

Of  trapeziums  there  are  two  kinds:  quad- 
rilaterals, that  have  two  only  of  the  sides  paral- 
lel, called  trapezoids;  and  quadrilaterals  that 
have  no  two  sides  parallel,  which  take  the  name 
of  trapeziums. 

77.  Give  a  sketch  of  a  square,  of  a  rhombus, 
of  a  rectangle,  of  a  rhomboid,  of  a  trapezoid, 
and  of  a  trapezium. 

The  line  that  joins  the  opposite  angles  of  a 
quadrilateral  is  called  a  diagonal. 

78.  Show  that  each  variety  of  quadrilateral 
has  two  diagonals,  and  say  in  which  kind  the 
diagonals  can  be  of  equal  lengths,  and  in  which 
they  cannot 

In  geometry,  one  figure  is  said  to  be  placed 
in  anotner,  when  the  inner  figure  is  wholly 


34  INVENTIONAL   GEOMETRY 

within  the  outer,  and  at  the  same  time  touches 
the  outer  in  as  many  points  as  the  respective 
forms  of  the  two  figures  will  admit. 

79.  Describe  a  circle  that  shall  have  a  diam- 
eter of  1J  inch,  and  place  a  square  in  it. 

80.  Can  you  make  a  rhombus  ? 

When  a  rhombus  has  its  obtuse  angles  twice 
the  size  of  those  which  are  acute,  it  is  called  a 
regular  rhombus. 

81.  Can  you  make  a  regular  rhombus  I 

82.  Can  you  make  a  rectangle  t  * 

83.  Can  you  make  a  rhomboid  ? 

84.  Can  you  make  a  trapezoid  ? 

85.  Can  you  make  a  trapezium  ? 

When  a  geometrical  figure  has  more  than 
four  sides,  it  takes  the  name  of  polygon,  which 
means  many-angled ;  and  when  a  polygon  has 
all  its  sides  equal,  and  all  its  angles  equal,  it  ife 
called  a  regular  polygon. 

A  polygon  that  has  five  sides  is  called  a  pen 
tagon. 

1  Rectangles  are  sometimes  called  oblongs,  and  sometimes 
long  squares. 


INVEXT10NAL   GEOMETRY.  .       35 

A  polygon  that  has  six  sides  is  called  a  hex- 
Agon. 

A  polygon  that  has  seven  sides  is  called  a 
heptagon. 

A  polygon  that  has  eight  sides  is  called  an 
octagon. 

A  polygon  that  has  nine  sides  is  called  a 
nonagon. 

A  polygon  that  has  ten  sides  is  called  a  dec- 
agon. 

A  polygon  that  has  eleven  sides  is  called  an 
undecagon. 

A  polygon  that  has  twelve  sides  is  called  a 
dodecagon. 

You  have  made  a  sector  with  a  reentrant 
angle. 

86.  Of  how  few  lines  can  you  make  a  figure 
with  a  reentrant  angle  ? 

87.  Of  how  few  sides  can  you  make  a  figure 
with  two  reentrant  angles  ? 

88.  Of  how  few  sides  can  you  construct  a 
tigure  with  three  reentrant  angles  \ 

89.  Show    hew    many     equilateral     trian 


36  INVENTION AL   GEOMETRY. 

gles  may  be  placed  around  one  point  to  touch 
it. 

90.  Canyon  divide  a  circle  into  six  equal 

sectors  ? 

A  sector  that  contains  a  sixth  part  of  a  cir- 
cle is  called  a  sextant. 

91.  Make  a  sextant,  and  write  upon  it  its 
name. 

92.  Construct  an  equilateral  triangle,  and 
write  in  each  angle  the  number  of  degrees  it 
contains. 

93.  Can  you  place  a  circle  in  a  semi-circle  J 

94.  Can  you  place  a  hexagon  in  a  circle  ? 

95.  Can  you  divide  a  circle  into  eight  equal 
sectors  ? 

A  sector  that  contains  the  eighth  part  of  a 
circle  is  called  an  octant. 

96.  Make  an  octant,  and  in  it  write  its  name, 
and  underneath  state  the  number  of  degrees 
that  the  angle  of  an  octant  contains. 

97.  Make  a  regular  octagon  in  a  circle. 
That  point  in  a  spare  which  is  equally  dia- 


tttVXXTlONAL   GEOMETRY.  .  & 

taut  from  the  sides  of  that  square,  and  also 
equally  distant  from  the  angular  points  of  that 
square,  is  called  the  centre  of  that  square  ? 

98.  Draw  a  line  an  inch  and  a  half  long,  and 
erect  a  square  upon  it,  and  find  the  centre  of  it. 

99.  Can  you  place  a  circle  in  a  square  ? 

100.  Place  three  circles  so  that  the  circum- 
ference of  each  may  rest  upon  the  centres  of 
the  other  two,  and  find  the  centre  of  the  curvili- 
near figure,  which  is  common  to  all  three  circles. 

That  point  in  an  equilateral  triangle  which 

is  equally  distant  from  each  side  of  the  triangle, 

and  equally  distant  from  each  of  the  angular 

'  points  of  the  triangle,  is  called  the  centre  of  the 

triangle. 

101.  Can  you  make  an  equilateral  triangle 
whose  sides  shall  be  two  inches,  and  find  the  cen- 
tre of  it? 

102.  Can  yon  place  a  circle  in  an  equilateral 
triangle?    , 

103.  Can  you  divide  an  equilateral  triangle 
into  six  parts  that  shall  be  equal  and  similar  f 


38  INVENT10NAL   OMOMBTRT. 

•-A.  104.  Can  you  divide  an  equilateral  triangle 
into  three  equal  and  similar  parts  ? 

—105.  What  is  the  greatest  number  of  angles 
that  can  be  made  with  four  lines  j 

106.  Make  a  hexagon,  and  place  a  trigon  OD 
the  outside  of  each  of  its  boundaries,  and  say 
what  the  figure  reminds  you  of. 

107.  Can  you,  any  more  ways  than  one,  di- 
vide a  hexagon  into  two  figures  that  shall  be 
equal  to  each  other,  and  similar  to  each  other  ? 

108.  Can  you  divide  a  circle  into  three  equal 
sectors  i 

109.  Can  you  fit  an  equilateral  triangle  in 
a  circle  ? 

110.  Draw  two  lines  cutting  each  other,  and 
show  what  is  meant  when  it  is  said  that  those 
angles  which  are  vertically  opposite  are  equal 
to  one  another. 

111.  Can  you  place  two  squares  so  that  one 
angle  of  one  square  may  vertically  touch  one 
angle  of  the  other  square  ? 

112.  Can  you  place  two  hexagons  BO  that 


INVE3TWNAL   GEOMETRY.  39 

one  angle  of  one  hexagon  may  touch  vertically 
one  angle  of  the  other  ? 

113.  Can  you  place  two  octagons  so  that  one 
angle  of  one  octagon  may  touch  vertically  one 
angle  of  the  other  \ 

You  have  divided  a  line  into  two  equal  parts. 

114.  Can  you  divide  a  line  into  four  equal 


115.  Make  a  scale  of  inches,  and  with  its 
assistance  make  a  rectangle  whose  length  shall 
be  3  and  breadth  2  inches. 

116.  Draw  a  line,  and  on  it,  side  by  side, 
construct  two  right-angled  triangles  that  shall 
be  exactly  alike,  and  whose  corresponding  sides 
shall  face  the  same  way. 

When  a  line  meets  a  circle  in  such  a  direction 
as  just  to  touch  it,  and  yet  on  being  produced 
goes  by  it  without  entering  it,  such  line  is  called 
a  tangent  to  the  circle. 

117.  Describe  a  circle,  and  draw  a  tangent 
to  it. 

The  tangent  to  a  circle,  at  a  particular  point 


40  INVENTIONAL   GEOMETRY. 

in  the  circumference  of  that  circle,  is  at  right 
angles  to  a  radius  drawn  to  -that  point.  And 
as  every  point  in  the  circumference  of  a  circle 
may  have  a  radius  drawn  to  it,  so  every  point 
in  the  circumference  of  a  circle  may  have  a  tan 
gent  drawn  from  it. 

118.  Can  you  draw  a  tangent  to  a  circle 
that  shall  touch  the  circumference  in  a  point 
given  ? 

119.  Given  a  circle,  and  a  tangent  to  that 
circle ;  it  is  required  to  find  the  point  in  the 
circumference  to  which  it  is  a  tangent. 

120.  Given  a  line,  and  a  point  in  that  line; 
it  is  required  to  find  the  centre  of  a  circle,  hav- 
ing a  diameter  of  one  inch,  the  circumference 
of  which  shall  touch  that  line  at  that  point. 

121.  Show  by  a  figure  how  many  equilateral 
triangles  may  be  placed  around  one  equilateral 
triangle  to  touch  it. 

122.  Divide  a  square  into  four  equal   and 
similar  figures  several  ways,  and  give  the  name 
to  each  variety 


INVENTIONAL   GEOMETRY.  41 

123.  Can  you  place  two  hexagons  so  that  one 
side  of  one  hexagon  may  coincide  with  one  side 
of  the  other  ? 

>124.  Can  you  divide  a  circle  into  twelve 
equal  sectors  ? 

125.  Can  you  place  two  octagons  so  that  one 
side  of  one  octagon  may  coincide  with  one  side 
of  the  other  ? 

You  have  divided  a  sector  into  two  equal 
sectors,  and  an  angle  into  two  equal  angles. 

126.  Can  you  divide  a  sector  into  four  equal 
sectors,  and  an  angle  into  four  equal  angles  ? 

127.  Can  you  make  a  rhombus,  whose  long 
diagonal  shall  be  twice  as  long  as  the  short 
one? 

128.  Can  you  make  a  regular  dodecagon  in 
a  circle  ? 

129.  Can  you  show  how  many  squares  may 
be  made  to  touch  at  one  point  ? 

You  recollect  that  plane  figure  that  has  the 
fewest  lines  possible  for  its  boundaries. 

130.  Of  how  few  plane  surfaces  can  you 
make  a  solid  body  ? 


42  1XVEXT10XAL   &EOMETR). 

A  body  which  has  four  plane,  equal,  and 
similar  surfaces,  is  called  a  tetrahedron. 

131.  Make  a  hollow  tetrahedron  of  one  piece 
of  cardboard,  and  show  on  paper  how  you  ar- 
range the  surfaces  to  fit  each  other,  and  give  a 
sketch  of  the  tetrahedron  when  made. 

You  know  how  to  fit  a  square  in  a  circle. 

132.  Can  you  fit  a  square  around  a  circle  \ 
When  two  triangles  have  the  angles  of  one 

respectively  equal  to  the  angles  of  the  other,  but 
the  sides  of  the  one  longer  or  shorter  respective- 
ly than  the  sides  of  the  other,  such  triangles., 
though  not  equal,  are  said  to  be  similar  each  to 
the  other.  Now  you  have  made  two  triangles 
that  are  equal  and  similar. 

133.  Can  you  make  two  triangles  that  shall 
not  be  equal,  and  yet  be  similar  ? 

134.  Make  a  rhomboid,  and  divide  it  several 
ways  into  two  figures  that  shall  be  equal  to  each 
other,  and  similar  to  each  other,  and  write  on 
each  figure  its  appropriate  name. 

135.  Make  two  equal  and  similar  rhomboids, 
and  divide  one  into  two  equal  and  similar  trian 


INVENT10NAL  GEOMETRY.  48 

gles  by  means  of  one  diagonal,  and  the  other 
into  two  equal  and  similar  triangles  by  means 
of  the  other  diagonal. 

136.  Can  yt>u  make  two  triangles  that  shall 
be  equal  to  each  other,  and  yet  not  similar  \ 

137.  Can  you  show  that  all  triangles  upon 
the  same  base  and  between  the  same  parallels 
are  equal  to  one  another  ? 

-138.  Can  you  place  a  circle,  whose  radius  is 
1J  inch,  so  that  its  circumference  may  touch 
two  points  4  inches  asunder  I 

139.  How  many  squares    may  be    placed 
around  one  square  to  touch  it  ? 

140.  Divide  a  rhombus  into  four  equal  and 
similar  figures  several  ways,  and  write  in  each 
figure  its  proper  name. 

v,  141.  Show  how  many  hexagons    may   be 
made  to  touch  one  point. 

142.  Show  how  many  circles  may  be  made 
to  touch  one  point  without  overlapping,  and 
compare  that  number  with  the  number  of  hexa- 
gons, the  number  of  squares,  and  the  number 

rf  equilateral  triangles. 


44  INVENTIONAL  GEOMETRY. 

When  a  body  has  six  equal  and  similar  stu 
faces  it  is  called  a  hexahedron. 

143.  Make  of  one  piece  of  card  a  holloa 
hexahedron.     Show  on  paper  how  you  arrange 
the  surfaces  so  as  to  fold  together,  and  give  a 
sketch  of  the  hexahedron  when  finished ;  and 
say  what  other  names  a  hexahedron  has. 

144.  Can  yon  make  a  right-angled  triangle, 
whose  base  shall  be  4  and  perpendicular  6  ? 

In  a  right-angled  triangle,  the  side  which 
faces  the  right  angle  is  called  the  hypothenuse. 

145.  Can  you  make  a  right-angled  triangle, 
whose  base  shall  be  4  and  hypothenuse  6  ? 

146.  Can    you    make    a   rectangle,   whose 
length  shall  be  5  and  diagonal  6  ? 

147.  Divide  a  rectangle  several  ways  into 
four  equal  and  similar  figures,  and  write  upon 
each  figure  its  proper  name. 

The  term  vertex  means  the  crown,  the  top, 
the  zenith ;  and  yet  the  angle  of  an  isosceles  tri 
angle  which  is  contained  by  the  equal  sides  is 
called  the  vertical  angle,  however  such  triangle 
may  be  placed;  and  the  side  opposite  to  suo 


INVENTIONAL   GEOMETRY.  45 

angle  is  still  called  the  base,  although  it  may 
not  happen  to  be  the  lowermost  side. 

148.  Place  in  different  positions  four  isos- 
?«les  triangles,  and  point  out  the  vertex  of  each. 

149.  Construct  an  isosceles  triangle,  whose 
base  shall  be  1  inch,  and  each  of  the  equal  sides 
2  inches,  and  place  on  the  opposite  side  of  the 
base  another  of  the  same  dimensions. 

150.  Can  you  invent  a  method  of  dividing 
a  circle  into  four  equal  and  similar  parts,  having 
other  boundaries  rather  than  the  radii  ? 

You  have  made  a  square,  and  placed  an 
equilateral  triangle  on  each  of  its  sides. 

151.  Can  you  make  an  equilateral  triangle, 
and  place  a  square  on  each  of  its  sides  ? 

152.  Can  you  fit  a  square  inside  a  circle,  and 
another  outside,  in  such  positions  with  regard  to 
each  other  as  shall  show  the  ratio  the  inner  one 
has  to  the  outer  ? 

153.  Can  you  divide  a  hexagon  into  four 
equal  and  similar  parts  ? 

154.  Can  you  divide  a  line  into  two  such 


^6  INVENTION AL   GEOMETRY . 

parts  that  one  part  shall  be  three  times  the 
length  of  the  other  ? 

155.  Can  you  divide  a  line  into  four  equal 
parts,  without  using  more  than  three  circles  ? 

156.  Can  you  make  a  triangle  whose  sides 
shall  be  2,  3,  and  4  inches  \ 

157.  Make  a  scale  having  the  end  division 
to  consist  of  ten  equal  parts  of  a  unit  of  the 
scale,  and  with  its  assistance  make  a  triangle 
whose  sides  shall  have  25,  IS,  and  12  parts  oi 
that  scale. 

158.  Can  you  construct  a  square  on  a  line 
without  using  any  other  radius  than  the  length 
of  that  line » 

159.  Can  you  make  a  circle  so  that  the  cen- 
tre may  not  be  marked,  and  tind  the  centre  by 
geometry  i 

_160.  Can  you  divide  an  equilateral  triangle 
into  four  equal  and  similar  parts  \ 

When  a  body  has  eight  surfaces,  whose  sides 
And  angles  are  all  respectively  equal,  it  is  called 
an  octahedron. 


INVENTIONAL  GEOMETRY.  47 

161.  Make  of  one  piece  of  card  a  hollow  oc- 
tahedron ;  show  how  you  arrange  the  surfaces 
so  as  to  fold  together  correctly;   and  give  a 
sketch  of  the  octahedron. 

162.  Can  you  divide  an  angle  into  four  equal 
angles,  without  using  more  than  four  circles  ? 

163.  In  how  many  ways  can  you  divide  an 
equilateral  triangle  into  three  parts,  that  shall 
be  equal  to  each  other,  and  similar  to  each 
other! 

164.  Given  an  arc  of  a  circle :  it  is  required 
to  find  the  centre  of  the  circle  of  which  it  is  an 
arc. 

165.  Can  you  make  a  symmetrical  trape- 
poidJ 

166.  Can  you  make  a  symmetrical  trape- 
zium) 

167.  Is  it  possible  to  make  a  rhomboid  with 
out  using  more  than  one  circle  ? 

168.  Is  it  possible  to  make  a  symmetrical 
trapezium,  using  no  more  than  one  circle  ? 

169.  Can  you  place  a  hexagon  in  an  equilat 


GEOMETRY. 

eral  triangle,  so  that  every  other  angle  of  the 
hexagon  may  touch  the  middle  of  a  side  of  the 
equilateral  triangle  ? 

170.  Can  you  construct  a  triangle,  whose 
sides  shall  be  4,  5,  and  9  inches  ? 

171.  Can  you  make  an  octagon,  with  one 
side  given  ? 

172.  Is  it  possible  that  any  triangle  can  be 
of  such  a  form  that,  when  divided  in  a  certain 
^ay  into  two  parts  equal  to  each  other,  such 
parts  shall  have  a  form  similar  to  that  of  the 
original  triangle  ? 

173.  Show  what  is  meant  when  it  is  said 
that  triangles  on  equal  bases,  in  the  same  line,  and 
having  the  same  vertex,  are  equal  in  surface. 

174.  Can  you  divide  an  isosceles  triangle 
into  two  triangles  that  shall  be  equal  to  each 
other,  but  that  shall  not  be  similar  to  each  oth- 
er? 

175.  Can  you  divide  an  equilateral  triangle 
into  two  figures  that  shall  have  equal  surfaces, 
but  no  similarity  in  form  ? 


INVENT10NAL   GEOMETRY.  49 

176.  Can   you  fit  an   equilateral   triangle 
about  a  circle  ? 

177.  Can  you  divide  an  equilateral  triangle 
into  four  triangles,  that  shall  be  equal  but  dis- 
similar \ 

178.  Group  together  seven  hexagons  so  that 
each  may  touch  the  adjoining  ones  vertically  at 
the  angles. 

.   179.  Make  an  octagon,  and  place  a  square 
on  each  of  its  sides. 

180.  Can  you  convert  a  square  into  a  rhom- 
boid? 

181.  Can  you  convert  a  square  into  a  rhom- 
bus? 

182.  Can   you   convert  a  rectangle  into  a 
rhomboid  ? 

183.  Can   you  convert  a  rectangle  into  a 
rhombus  ? 

184.  Can  you  divide  any  triangle  into  fouf 
equal  and  similar  triangles  ? 

--185.  Can  you  invent  a  method  of  dividing  a 
line  into  three  equal  parts  H 


50  INVENTIONAL   &EOMETR*. 

186.  Can  you  place  a  hexagon  in  an  equilat 
eral  triangle,  so  that  every  other  side  of  the 
hexagon  may  touch  a  side  of  the  triangle  \ 

187.  Can  you  divide  a  line  into  two  such 
parts  that  one  part  may  be  twice  the  length  of 
the  other  I 

188.  Can  you  divide  a  rectangular  piece  ol 
paper  into  three  equal  strips*  by  one  cut  of  a 
knife  or  pair  of  scissors  ? 

189.  You  have  made  one  triangle  similar  to 
another,  but  not  equal ;  can  you  make  one  rec- 
tangle similar  to  another,  but  not  equal  ? 

190.  Can  you  make  a  square,  and  place  four 
octagons  round  it  in  such  a  manner  that  each 
side  of  the  square  may  form  one  side  of  one  of 
the  octagons  ? 

191.  Can  you  make  two  rhomboids  that  shall 
be  similar,  but  not  equal  ? 

192.  Can  you  place  a  circle,  whose  radius  is 
li  inch,  so  as  to  touch   two  points  2   inchei 
asunder  ? 

193    Can  you  place  an  octagon  in  a  square 


INVENTIONAL   GEOMETRY  51 

in  such  a  position  that  every  other  side  of  the 
octagon  may  coincide  with  a  side  of  the  square  ? 

194.  Fit  an  equilateral  triangle  inside  a  cir- 
cle, and  another  outside,  in  such  positions  with 
regard  to  each  other  as  shall  show  the  ratio  the 
inner  one  has  to  the  outer. 

195.  Can  you  place  four  octagons  in  a  group 
to  touch  at  their  angles  ? 

196.  Can  you  fit  a  hexagon  outside  a  circle  ? 

197.  Can  you  place  four  octagons  to  meet  in 
one  point,  and  to  overlap  each  other  to  an  equal 
extent  J 

198.  Can  you  let  fall  a  perpendicular  to  a  line 
from  a  point  given  above  that  line  ? 

Those  instruments  by  which  an  angle  can  be 
constructed  so  as  to  contain  a  certain  number  of 
degrees,  or  by  which  we  can  measure  an  angle, 
and  determine  how  many  degrees  it  contains,  as 
also  by  which  we  can  make  an  arc  of  a  circle 
that  shall  subtend  a  certain  number  of  degrees, 
or  can  measure  an  arc  and  determine  how  many 
degrees  it  subtends,  are  called  protractors. 


52  INVENT10NAL   GEOMETRY. 

Protractors  commonly  extend  to  180° ; 
though  there  are  protractors  that  include  the 
whole  circle,  that  is,  which  extend  to  360°. 


199.  Make  of  a  piece  of  card  as  accurate  a 
protractor  as  you  can. 

200.  Make  by  a  protractor  an  angle  of  45°, 
and  prove  by  geometry  whether  it  is  accurate 
or  not. 

:^201.  Can  you  contrive  to  divide  a  square 

into  two  equal  but  dissimilar  parts  ? 

202.  Make  with  a    protractor   an   angle  of 
60°,  and  prove  by  geometry  whether  it  is  cor- 
rect or  not. 

203.  Make  an  angle,  and  determine  by  the 
protractor  the  number  of  degrees  it  contains. 

204.  Make  by  geometry  the  arc  of  a  quad- 


GEOMETRY.  $3 

rant,  and  determine  by  the  protractor  the  num- 
ber of  degrees  that  arc  subtends. 

205.  Show  how  many  hexagons  may  be  made 
to  touch  one  hexagon  at  the  sides. 

That  which  an  angle  lacks  of  a  right  angle, 
that  is,  of  90°,  is  called  its  complement. 

206.  Make  a  few  angles,  and  say  which  their 
complements  are. 

207.  Make  an  angle  of  70°,  and  measure  its 
complement. 

That  which  an  angle  lacks  of  180°  is  called 
its  supplement. 

208.  Make  a  few  angles,  and  their  supple- 
ments, and  measure  them  by  the  protractor. 

209.  Make  by  geometry  an  angle  of  30°,  and 
its  supplement,  and  measure  by  the  protractor 
the  correctness  of  each. 

_^  210.  Can  you  make  a  semicircle  equal  to  a 
circle  ? 

^  211 .  Make  a  few  triangles  of  different  forms, 
and  measure  by  the  protractor  the  angles  of  each, 
and  see  if  you  can  find  a  triangle  whose  angles 


54  INTENTIONAL 

added  together  amount  to  more  than  the  angles 
of  any  other  triangle  added  together. 

212.  Can  you  make  a  pentagon  in  a  circle 
by  means  of  the  protractor  ? 

213.  Make  of  one  piece  of  card  a  hollow 
square  pyramid,  and  let  the  slant  height  be 
twice  the  diagonal  of  the  base.     Give  a  plan  of 
your  method,  and  a  sketch  of  the  pyramid, 
when  completed. 

214:.  Can  you  make  a  pentagon  outside  a 
circle  by  means  of  a  prgtractor  8 

215.  Can  you,  by  means  of  a  protractor, 
make  a  pentagon  without  using  a  circle  at  all  ? 

It  has  already  been  said  that  the  chord  of  an 
arc  is  a  line  joining  the  extremities  of  that  arc. 

216.  With  the  assistance  of  a  semicircular 
protractor,  can  you  contrive  to  place  on  one 
line  the  chords  of  all  the  degrees  from  1°  to 
90°?  or,  in  other  words,  can  you  make  a  line 
of  chords  ? 

217.  Can  you  say  why  the  line  of  chord? 
should  not  extend  as  far  as  180°  ? 


INVENTIONAL  GEOMETRY.  65 

There  is  one  chord  which  is  equal  in  lengtii 
to  the  radius  of  the  quadrant  to  which  all  the 
chords  belong;  that  is,  which  is  equal  to  the 
radius  of  the  line  of  chords. 

218.  Say  which  chord  is  equal  to  the  radius 
of  the  line  of  chords. 

219.  Make,  by  the  line  of  chords,  angles  of 
26°,  32°,  75°,  and  prove,  by  the  protractor, 
whether  they  are  correct  or  not. 

220.  How,  by  the  line  of  chords,  will  you 
make  an  obtuse  angle,  say  one  of  115°  ? 

221.  Can  you  make,  with  the  assistance  of 
a  line  of  chords,  a  triangle  whose  angles  at  the 
base  shall  each  be  double  of  the  angle  at  the 
vertex  ? 

222.  Make  a  triangle,  whose  sides  shall  be 
21,  15,  and  12,  and  measure  its  angles  by  the 
line  of  chords  and  by  the  protractor. 

,^  223.  There  is  one  side  of  a  right-angled  tri- 
angle that  is  longer  than  either  of  the  other 
two.  Give  its  name,  and  show  from  such  fact 
that  the  chord  of  45°  is  longer  than  half  the 
chord  of  90°. 


65  INVENT10NAL   GEOMETRY. 

224.  Make  by  the  piotractor  an  angle  of 
90°,  and  give  a  figure  to  show  which  you  con 
aider  the  most  convenient  way  of  holding  the 
protractor,  when,  to  a  line,  you  wish  to  raise  or 
let  fall  a  perpendicular. 

225.  Can  you  make  an  isosceles  triangle, 
having  its  base  1,  and  the  sum  of  the  other  two 
sides  3} 

226.  Can  you  determine,  by  means  of  the 
scale,  the  length  of  the  hy^othenuse  of  a  right- 
angled  triangle,  whose  base  is  4,  and  perpen- 
dicular 3  ? 

227.  Place  a  hexagon  inside  a  circle,  and 
another  outside,  in  such  positions  with  regard 
to  each  other  as  to  show  the  ratio  the  inner  one 
has  to  the  outer. 

By  the  area  of  a  figure  is  meant  its  super 
ficial  contents,  as  expressed  in  the  terms  of  any 
specified  system  of  measures. 

In  England,  the  system  of  linear  measures 
squared  is  generally  *  used  to  express  areas ;  a* 

1  The  terms  acres  and  roods  are  the  exceptions. 


INVENTWNAL   GEOMETRY.  57 

square  inches,  square  feet,  square  yards,  square 
poles,  square  chains,  square  miles. 

The  area  of  a  square  whose  side  is  one  inch 
IB  called  a  square  inch ;  and  a  square  inch  is  the 
unit  by  a  certain  number  of  which  the  areas  of 
all  squares  are  either  expressed  or  implied. 

The  area  of  a  square  in  square  inches  may 
be  found  by  multiplying  its  length  in  inches  by 
its  breadth,  or,  which  is  the  same  thing,  its  base 
by  its  perpendicular  height;  and  as,  in  the 
square,  the  base  and  perpendicular  height  are 
always  of  equal  extent,  the  area  of  a  square  is 
said  to  be  found  by  multiplying  the  base  by  a 
number  equal  to  itself,  that  is,  by  squaring  the 
base. 

228.  Make  squares  whose  sides  shall  repre- 
sent respectively,  1,  2,  3,  4,  5,  etc.,  inches,  and 
show  that  their  areas  shall  represent  respective- 
ly, 1,  4,  9,  16,  25,  etc.,  square  inches ;  that  is, 
shall  represent  respectively  a  number  of  inches 
that  shall  be  equal  to  T,  2s,  3*,  49,  59,  etc. 

229.  Make  equilateral  triangles,  whose  sides 
shall  represent  1,  2,  3,  4,  5,  etc.,  inches,  respee 


Jg  INVENT10NAL   GEOMETRY. 

lively,  and  show  that  their  areas  (though  not 
actually  so  much  as  1,  4,  9,  16,  25,  etc.)  are  in 
the  ratio  of  1,  4,  9,  16,  25,  etc. ;  that  is,  that 
their  areas  are  in  the  ratio  of  the  squares  of 
their  sides. 

230.  How   would   you  express  in  general 
terms  the  relation  existing  between  the  sides 
and  areas  of  similar  figures  ? 

231.  Show  by  a  figure  that  a  square  yard 
contains  9  square  feet ;  that  is,  that  the  area  of 
a  square  yard  is  equal  to  9  square  feet. 

232.  Give  a  figure  of  half  a  square  yard,  and 
another  of  half  a  yard  square,  and  say  what  re- 
lation one  bears  to  the  other. 

233.  Show  that  the  area  of  a.  square  foot  is 
equal  to  144  square  inches. 

234.  Can  you  show  that  the  squares  upon 
the  two  sides  of  a  right-angled  isosceles  triangle 
are  together  equal  to  the  square  upon  the  hy- 
po thenuse  ? 

Geometricians  have  demonstrated  that  a  tri 


INVENTION AL   GEOMETRY  59 

augle,  whose  sides  are  3,  4,  and  5,  .a  a  right- 
angled  triangle. 

235.  Make  a  triangle,  whose  sides  are  3,  4, 
and  5 ;  erect  a  square  on  each  of  such  sides,  and 
see  how  any  two  of  the  squares  are  related  to 
the  third  square. 

_:>236.  Can  you  raise  a  perpendicular  to  a  line, 
and  from  the  end  of  it  3 

237.  Can  you  find  other  three  numbers,  be- 
sides 3,  4,  and  5,  such  that  the  squares  of  the 
less  two  numbers  shall  together  be  equal  to  the 
square  of  the  greater,  and  show  that  the  trian- 
gles they  make,  so  far  as  the  eye  can  judge,  by 
the  assistance  of  a  protractor,  are  right-angled 
triangles  ? 

The  area  of  a  rectangle,  whose  base  is  4,  and 
perpendicular  3,  is  12. 

238.  Show  by  a  figure  that  the  area  of  & 
right-angled  triangle,  whose  base  is  4,  and  per- 
pendicular 3,  is  half  4x3;  i.  e.,  is  ~j-  =  j  =  6- 

A  solid  bounded  by  six  rectangles,  having 


50  INVENTIONAL 

only   the  opposite  ones  similar,   parallel  and 
equal,  is  called  a  parallelepiped. 

The  most  common  dimensions  of  the  paral- 
lelopiped  called  a  building-brick  are  9,  4J,  and 
3  inches. 

239.  Make  of  one  piece  of  cardboard  a  paral 
lelopiped  of  the  same  form  as  a  common  build- 
ing-brick ; l  show  how  you  arrange  all  the  sides 
to  fit,  and  give  a  sketch  of  it. 

It  is  now  above  2.000  years  since  geometri- 
cians discovered  that  the  square  upon  the  base 
of  any  right-angled  triangle,  together  with  the 
square  upon  the  perpendicular,  is  equal  to  the 
square  upon  the  hypothenuse. 

You  have  proved  that  the  squares  upon  the 
two  sides  of  a  right-angled  isosceles  triangle  are 
together  equal  to  the  square  upon  the  hypothe 
nuse. 

240.  Can  you  invent  any  method  of  proving 
to  the  eye  that  the  squares  upon  the  base  and 
perpendicular  of  any  right-angled  triangle  what 

1  When  a  parallelepiped  is  long,  it  takes  the  name  of  bar 
u  a  bar  of  iron. 


INVENTIONAL   GEOMETRY.  61 

ever  are  together  equal  to  the  square  upon  the 
hypothenuse  \ 

241.  Construct  a  triangle,  whose  base  shall 
be  12,  and  the  sum  of  the  other  two  sides  15, 
and  of  which  one  side  shall  be  twice  the  length 
of  the  other. 

242.  Can  you  make  one  square  that  shall  be 
equal  to  the  sum  of  two  other  squares  ? 

243.  Can  you  make  a  square  that  shall  equal 
the  difference  between  two  squares ! 

244.  Can  you  make  a  square  that  shall  equal 
in  surface  the  sum  of  three  squares. 

The  angle  made  by  the  two  lines  joining  the 
centre  of  a  polygon  with  the  extremities  of  one 
of  its  sides  is  called  the  angle  at  the  centre  of 
the  polygon  ;  and  the  angle  made  by  any  two 
contiguous  sides  of  a  polygon  is  called  the  angle 
of  the  polygon. 

245.  Make  an  octagon  in  a  circle,  measure 
oy  a  line  of  chords  the  angle  at  the  centre  and 
the  angle  of  the  octagon,  and  prove  the  correct- 
ness of  your  work  by  calculation. 


62  INVENTIONAL  GEOMETRY. 

A  scale  having  its  breadth  divided  into  ten 
equally  long  and  narrow  parallel  spaces,  cut  at 
equal  intervals  by  lines  at  right  angles  to  them, 
with  a  spare  end  division  subdivided  similarly, 
only  at  right  angles  to  the  other  divisions,  into 
ten  small  rectangles,  each  of  which  small  rec- 
tangles, being  provided  with  a  diagonal,  is  called 
a  diagonal  scale. 

246.  Make  a  diagonal  scale  that  shall  express 
a  number  consisting  of  three  digits. 

247.  With  the  assistance  of  a  diagonal  scale, 
construct  a  plan  of  a  rectangular  piece  of  ground, 
whose  length  is  556  yards,  and  breadth  196 
yards,  and  divide  it  by  lines  parallel  to  either  end 
into  four  equal  and  similar  gardens,  and  name 
the  area  of  the  whole  piece  and  of  each  garden. 

When  a  pyramid  is  divided  into  two  parts 
by  a  plane  parallel  to  the  base,  that  part  next 
the  base  is  called  a  frustum  of  that  pyramid. 

248.  Make  of  one  piece  of  card  the  frustun 
of  a  pentagonal  pyramid,  and  let  the  small  end 
of  the  frustum  contain  one-half  the  surface  of 
that  which  the  greater  end  contains. 


USVEETIONAL   GEOMETRY.  63 

249.  Out  of  a  piece  of  paper,  having  irregu- 
lar boundaries  to  begin  with,  make  a  square, 
using  no  instruments  besides  the  fingers. 

250.  Can  you  show  by  a  figure  in  what  casea 
the  square  of  £  is  of  the  same  value  as  £  of  i, 
and  in  what  cases  the  square  of  •£  is  of  greater 
value  than  J  of  £  ? 

251.  Construct,  by  a  diagonal  scale,  a  trian- 
gle whose  three  sides  shall  be  equal  to  791,  489, 
and  568. 

252.  Can  you  show  to  the  eye  how  much  4 
is  greater  than  \  ? 

253.  How  many   ways  can  you  show  of 
drawing  one  line  parallel  to  another  line,  and 
through  a  given  point  \ 

254.  Show  by  a  figure  how  many  square 
inches  there  are  in  a  square  whose  side  is  1-J 
inch,   and  prove   the  truth  of  the  result  by 
arithmetic. 

255.  Show  by  a  figure  how  many  square 
yards  there  are  in  a  square  pole. 

You  know  how  to  find  the  area  of  a  rectan 


64  INTENTIONAL    GEOMETRY. 

gle,  and  you  have  changed  a  rectangle  into  a 
rhomboid. 

256.  How  would  you  find  the  area  of  ft 
rhombus  ? 

257.  Can  you  make  a  right-angled  isosceles 
triangle  equal  to  a  square  ? 

258.  Can  you  make  a  circle  half  the  size  of 
another  circle  ? 

259.  Can  you  make  an  equilateral  triangle 
double  the  size  of  another  equilateral  triangle  ? 

260.  Make  of  one  piece  of  cardboard  a  hol- 
low rhombic  prism  ;  show  how  you  arrange  the 
sides  to  fit ;  and  give  a  sketch  of  the  prism  when 
complete. 

261.  Make    a    square,   whose    length    and 
breadth   are   6,   and    make   rectangles,   whose 
lengths  and  breadths  are  7  and  5,  8  and  4,  9 
and  3,  10  and  2,  and  11  and  1,  and  show  that, 
though  the  sums  of  the  sides  are  all  equal,  the 
areas  are  not  all  equal. 

262.  What  is  the  largest  rectangle  that  ran 
je  placed  in  an  isosceles  triangle  f 


INVEXTIONAL   GEOMETRY.  t)5 

263.  Show  by  a  figure  which  is  greater,  and 
how  much,  2  solid  inches  or  2  inches  solid. 

If  from  one  extremity  of  an  arc  there  be  a 
line  drawn  at  right  angles  to  a  radius  joining 
that  extremity,  and  produced  until  it  is  inter- 
cepted by  a  prolonged  radius  passing  through 
the  other  extremity,  such  line  is  called  the  tan- 
gent of  that  arc. 

You  have  given  an  example  of  a  tangent  to 
a  circle. 

264.  Give  an  example  of  a  tangent  to  an  arc. 

265.  Can  you  draw  a  tangent  to  an  arc  of 
90°  t 

266.  Can  you  contrive  to  place  on  one  line 
the  tangents  to  the  arcs  of  all  the  degrees,  from 
that  of  one  degree  to  that  of  about  85° ;  i.  e., 
can  you  make  a  line  of  tangents  ? 

267.  Show   which   tangent,  or  rather,  the 
tangent  to  which  arc,  is  equal  to  the  radius  of 
the  line  of  tangents. 

268.  Make,  by  the  line  of  tangents,  angles 
of  20°,  40°,  75°,  and  80°. 


66  IN7ENTIONAL   GEOMETRY. 

That  solid  whose  faces  are  six  equal  and 
regular  rhombuses  is  called  a  regular  rhoinbo- 
hedron. 

269.  Make  in  card  a  regular  rhombohedron, 
show  how  the  sides  are  adjusted  to  fit,  and  give 
a  sketch  of  it  when  made. 

A  tangent  to  the  complement  of  an  arc  ia 
called  the  complement  tangent,  or  the  co-tan 
gent. 

270.  Make  a  few  arcs,  and  their,  tangents, 
and  their  co-tangents. 

271.  Make  an  angle,-  and  its  tangent,  and 
also  its  co-tangent. 

272.  Can  you  make  an  angle  of  130°  by  the 
line  of  tangents  ? 

273.  Can  you  find  out  a  method  of  making 
an  angle  of  90°  by  the  line  of  tangents  ? 

274.  Measure  a  few  acute  angles  by  the  line 
af  tangents. 

275.  Measure  an  obtuse  angle  by  the  line  of 
tangents. 

276.  Can    you    make    a    rectangle,    whose 


INTENTIONAL   GEOMETRY.  67 

length  is  9,  and  breadth  4,  and  divide  it  into 
two  parts  of  such  a  form  that,  being  placed  to 
touch  in  a  certain  way,  they  shall  make  a 
square  i 

277.  Show  that  the  area  of  a  trapezium  may 
be  found  by  dividing  th6  trapezium  into  two 
triangles  by  a  diagonal,  and  finding  the  sum  of 
the  areas  of  such  triangles. 

278.  Make  a   square,  whose  side  shall  be 
one-third  of  a  foot,  and  show  what  part  of  a 
foot  it  contains,  and  how  many  square  inches. 

279.  Can  you,  out  of  one  piece  of  card,  make 
a  truncated  tetrahedron,  and  show  how  you  ar- 
range the  sides  to  fit,  and  give  a  sketch  of  it 
when  made  \ 

280.  Can  you  make  a  hexagon,  whose  sides 
ghall  all  be  equal,  but  whose  angles  shall  not  all 
r>e  equal,  and  that  shall  yet  be  symmetrical  ? 

281.  Can  you  make  a  right-angled  trapezoid 
equal  to  a  square  ? 

282.  Can  you  make  a  circle  three  times  as 
large  as  another  circle  t 


58  INTENTIONAL   GEOMETRY. 

283.  Make  by  the   protractor   a  nonagon, 
whose  sides  shall  be  half  an  inch,  and  measure 
the  angles  of  the  nonagon  by  the  line  of  tan- 
gents. 

284.  How  many  dodecagons  may  be  made 
to  touch  one  dodecagon  at  the  angles  ? 

285.  How  many  dodecagons  may  be  made 
to  touch  one  dodecagon  at  the  sides  ? 

286.  Show  by  a  figure  how  many  bricks  of 
9  inches  by  4^,  laid  flat,  it  will  take  to  cover  a 
square  yard,  and  prove  it  by  calculation. 

287.  Can  you   determine   the  number  of 
bricks  it  would  take  to  cover  a  floor,  6  yards 
long  and  5£  wide,  allowing  50  for  breakage  ? 

288.  How  would  you  make  a  square  by 
means  of  the  protractor  and  a  pencil,  without  a 
pair  of  compasses  ? 

289.  Can  you  bisect  an  angle  without  using 
circles  or  arcs  ? 

290.  Construct  of  one  piece  of  card  a  hollow 
truncated  cube  ;  show  on  paper  bow  you  arrange 


INVENTIONAL    GEOMETRY.  69 

the  sides  to  touch,  and  give  a  sketch  of  the  trun- 
cated cube  when  made. 

291.  Can  you  make  a  pentagon,  whose  side 
ahaL  be  one  inch,  without  using  a  circle,  and 
without  having  access  to  the  centre  of  the  pen- 
tagon ? 

292.  Can  you  pass  the  circumference  of  a 
circle  through  the  angular  points  of  a  triangle  ? 

293.  Show  how  you  would  find  the  area  of 
a  reentrant-angled  trapezium. 

294.  Exhibit  to  the  eye  that  i  +  i  + 1  =  1. 

295.  Place  a  circle  about  a  quadrant. 

If  to  one  extremity  of  an  arc,  not  greater 
than  that  of  a  quadrant,  there  be  drawn  a 
radius,  and  if  from  the  other  extremity  there 
be  let  fall  a  perpendicular  to  that  radius,  such 
perpendicular  is  called  a  sine  of  that  arc. 

296.  Make  a  few  arcs  of  circles  and  their 
lines. 

297.  Can  you  place  a  circle  in  a  triangle! 
2%.  Can  you  contrive  to  place  on  one  line 


70  INVEXTIOXAL   GEOMETKY. 

the  sines  of  all  the  degrees  from  1°  to  90°  8  in 
other  words,  can  you  make  a  line  of  sines  ? 

299.  Say   which   of   the  sines  is  equal  ID 
tength  to  the  radius  of  the  line  of  sines. 

300.  Given  the  perpendicular  of  an  equilat- 
eral triangle,  to  construct  that  equilateral  tri- 
angle. 

When  a  body  has  twelve  equal  and  similar 
surfaces,  it  is  called  a  dodecahedron. 

301.  Make  of  one  piece  of  card  a  hollow 
dodecahedron ;  show  on  paper  how  you  arrange 
the  surfaces  to  fit,  and  give  a  sketch  of  the  do- 
decahedron when  made. 

302.  Measure  by  the  line  of  sines  a  few 
acute  angles. 

303.  Can  you  make  an  angle  of  70°  by  the 
line  of  sines  ? 

The  sine  of  the  complement  of  an  arc  it 
called  the  co-sine  of  that  arc. 

304.  Show  by  a  figure  that  the  co-sine  of 
the  arc  of  35°  is  equal  to  the  sine  of  55°. 

305.  Given  alone  the  distance  between  the 


INVENTIONAL   GEOMETRY.  71 

parallel  sides  of  a  regular  hexagon,  to  construct 
that  hexagon. 

306.  Fit  a  segment  of  a  circle  in  a  rectangle 
whose  length  is  3  and  breadth  1. 

307.  Can  you  fit  a  segment  of  a  circle  in  a 
rectangle  whose  length  is  3  and  breadth  2  ? 

308.  Can  you  place  a  circle  in  a  quadrant  ? 

309.  Give  a  figure  of  a  symmetrical  trape- 
zoid  whose  parallel  sides  are  40  and  20,  and 
the  perpendicular  distance  between  them  60; 
measure  its  angles  by  the  line  of  sines,  and  cal- 
culate the  area. 

310.  Show  by  a  figure  what  the  ,area  of  a 
rectangle  is,  whose  length  is  2£  and  breadth  1-J, 
and  prove  it  by  calculation. 

311.  Given,  from  a  line  of  chords,  the  chord 
of  90°,  it  is  required  to  find  the  radius  of  that 
line  of  chords. 

You  have  drawn  one  triangle  similar  to 
another,  and  one  rhomboid  similar  to  another  ; 
car  you  draw  one  trapezium  similar  to  another  ? 

H12.  Make  of  one  pioce  of  card  a  hollow  ein 


72  INVENTIONAL   GEOMETRY. 

bossed  tetrahedron  ;  show  how  you  arrange  the 
surfaces  to  fit,  and  give  a  sketch  of  it  when  com- 
pleted ;  and  say  if  you  can  so  arrange  the  sur- 
faces on  a  plane  as  to  have  no  reentrant  an 
gles. 

313.  Can  you  make  one  triangle  similar  to 
another,  and  twice  the  size  ? 

314.  Can  you  make  an  irregular  polygon 
similar  to  another,  and  twice  the  size  i 

315.  Can  you  make  an   irregular  polygon 
similar  to  another,  and  half  the  size  ? 

316.  Can  you  change  a  square  to  &n  obtuse- 
angled  isosceles  triangle  ? 

317.  Can  you  show  by  a  figure  how  much 
more  £  is  than  \  ? 

318.  Can  you  make  an   isosceles   triangle, 
each  of  whose  sides  shall  be  half  the  base  ? 

319.  Can  you  determine  the  size  of  an  ob- 
tuse angle  by  the  line   of  sines  ? 

320.  Can  you  show  by  a  figure  that  2  is  con 
tained  in  3  \\  time  ? 


INVENTION AL   GEOMETRY.  73 

321.  Can  you  «how  that  the  sine  of  an  arc 
is  half  the  chord  of  double  the  arc  ? 

322.  Take  an  inch  to  represent  a  foot,  and 
make  a  scale  of  feet  and  inches. 

323.  From  the  theorem,  that  triangles  on 
the  same  base,  and  between  the  same  parallels^ 
are  equal  in  surface,  can  you  change  a  trapezi 
um  into  a  triangle  ? 

324.  Can  you  change  a  triangle  into  a  rec- 
tangle \ 

325.  Make  of  a  piece  of  card  a  hexahedron, 
embossed  with  semi-octahedrons;  give  a  plan 
of  the  method  by  which  you  arrange  the  sur 
faces  to  fit,  and  give  a  sketch  of  the  figure  when 
made. 

326.  Can  you  convert  a  common  trapezium 
into  a  symmetrical  trapezium  ? 

327.  Can  you  construct  a  square,  whose  di- 
agonal shall  be  3  inches,  and  find  the  area  of 
it? 

That  portion  of  the  radius  ot  an  arc  which 
is  intercepted  between  the  $ine  and  the  extrem 


74  INTENTIONAL   GEOMETRY. 

ity  of  the  arc  is  called  the  versed  sine  of  that 
arc. 

328.  Give  an  example  of  the  versed  sine  of 
an  arc. 

329.  Beginning  at  a  point  in  a  line,  can  you 
arrange  the  versed  sines  of  all  the  degrees  from 
1°  to  90°  ?  i.  e.,  can  you  make  a  line  of  versed 
sines! 

330.  Show  when  the  versed  sine  of  an  arc 
is  equal  to  the  sine  of  the  arc. 

331.  Show  when  the  versed  sine  of  an  arc  is 
equal  to  half  the  chord  of  the  arc. 

332.  Say  what  versed  sine  is  equal  to  the 
radius  of  the  quadrant  to  which  the  line  of 
versed  sines  belongs. 

333.  Given  the  versed  sine  of  an  arc  equal 
to  half  the  radius  of  that  arc,  to  determine  the 
number  of  degrees  in  that  arc. 

334.  Can  you  reduce 'a  figure  of  five  sides  to 
.  ft  triangle  and  to  a  rectangle  ? 

When  lines  or  curves,  or  both,  are  syminet 


INVENTIONAL   GEOMETRY.  75 

rically  grouped  about  a  point  for  effect,  thev 
take  the  name  of  star. 

335.  Invent  and   construct   as  beautiful    a 
star  as  you  can. 

When  a  body  has  20  surfaces,  whose  side* 
and  angles  are  respectively  equal,  it  is  called  an 
icosahedron. 

336.  Make  of  one  piece  of  card  a  hollow 
icosahedron ;  *  represent  on  paper  the  method 
by  which  you  arrange  the  surfaces  to  fit,  and 
give  a  sketch  of  the  icosahedron  when  made. 

337.  Describe  an  arc ;  let  it  be  less  than  that 
of  a  quadrant,  and  draw  to  it  the  chord,  the 
tangent,  and  co-tangent,  the  sine,  and  •  co-sine, 
and  the  versed  sine. 

338.  Given  the  sine  of  an  arc,  exactly  one- 
tburth  of  the  radius  of  that  arc ;  it  is  required, 
by  the  protractor,  to  determine  in  degrees  the 
length  of  such  arc. 

1  The  tetrahedron,  the  hexahedron,  the  octahedron,  tht 
dodecahedron,  and  the  icosahedron,  take  the  name  of  regular 
oodles.  These  five  regular  bodies  are  a<so  called  Platonic 
bodies  and  along  with  these  Platonic  bodies  some  place  tbf 
sphere,  as  the  most  regular  of  all  bodies. 


76  INVENTIONAL  GEOMETRY. 

339.  Given  the  versed  sine  of  an  arc,  exactly 
one-fourth  of  the  radius  of  that  arc;  it  is  re 
quired,  by  the  protractor,  to  determine  the  de 
in  that  arc. 


340.  How  would  you  prove  the  correctness 
of  a  straight-edge,  of  a  parallel  ruler,  of  a  set 
square,  of  a  drawing-board,  of  a  protractor,  and 
of  a  line  of  chords  ? 

341.  Reduce  an  irregular  hexagon  with  a  re- 
entrant angle  to  a  triangle. 

342.  Reduce  an  irregular  octagon  with  two 
reentrant  angles  to  a  triangle. 

It  has  been  agreed  upon  by  arithmeticians 
that  fractions  whose  denominators  are  either  10, 
or  some  multiple  of  10,  as  -j^-,  -fff,  ^V^/V,  etc., 
may  be  expressed  without  their  denominators, 
by  placing  a  dot  at  the  left  hand  of  the  numer- 
ator: thus,  ^  may  be  expressed  .5  ;  -ffo  thus, 
25  ;  jfifa  thus,  .125  ;  and  ^  thus,  .05. 

Such  expressions  are  called  decimals. 

Like  other  fractions,  decimals  may  be  illus- 
trated either  by  a  line  and  parts  of  that  line,  01 
by  a  surfece  and  parts  of  that  surface 


INVENTIONAL    GEOMETRY.  77 

343.  By  dividing  a  line,  supposed  to  repre- 
sent a  unit  of  length,  illustrate  the  value  of  .5, 
.25,  and  .125,  etc. 

344.  By  means  of  a  square  representing  a 
unit  of  surface,  exhibit  the  value  of  .5,  .25,  and 
125. 

345.  Out  of  an  apple,  or  a  turnip,  or  a  pota- 
to, cut  a  cube :  call  each  of  its  linear  dimensions 
2,  and  determine  its  solid  content,  and  prove 
by  arithmetic. 

346.  Show  by  means  of  a  cube,  and  prove 
by  arithmetic,  what  the  cube  of  1  £  is. 

347.  Can  you  place  nine  trees  in  ten  rows 
of  three  in  a  row  ? 

348.  With  10  divisions  of  a  diagonal  scale 
for  its  side,  construct  an  equilateral  triangle, 
and  call  such  side  1 ;  and  determine  the  length 
of  its  perpendicular  to  three  decimal  places,  and 
prove  its  truth  by  calculation. 

349.  Can  you  calculate  the  area  of  an  equi 
lateral  triangle  whose  side  is  1 J 


78  IXVEXTIONAL   GEOMKTRF. 

350.  Illustrate  by  geometry  the  respective 
values  of  .9,  .99,  .999,  .9999. 

A  circle  may  be  supposed  to  consist  of  an 
indefinite  number  of  equal  isosceles  triangles, 
Slaving  their  "bases  placed  along  the  circumfer- 
ence of  the  circle,  and  their  vertices  all  meeting 
in  the  centre  of  the  circle.  And  as  the  areas  of 
all  these  triangles  added  together  would  be 
equal  to  the  area  of  the  circle : 

To  find  the  area  of  a  circle — multiply  the 
radius  which  is  the  perpendicular  common  to  all 
these  imaginary  triangles,  by  the  circumference 
which  is  the  sum  of  all  their  bases,  and  divide 
the  product  by  2. 

Reckoning  the  circumference  of  a  circle  aa 
3^  times  its  diameter : 

351.  Find  the  area  of  a  circle  whose  diam 
eter  is  1. 

Reckoning  the  circumference  of  a  circle  tu 
be  3.M16  times  the  diameter: 

352.  Find  the  area  of  a  circle  whose  diam 
is  1. 

Circles  being  eimilar  figures,  the  areas  of 


IM9M&T10NAL   GEOMETRY.  79 

circles  are  to  each  other  as  the  squares  of  their 
radii,  their  diameters,  or  their  circumferences. 

353.  Find  the  area  of  a  circle  whose  radius 
is  5,  and  find  the  area  of  another  circle  whose 
radius  is  7,  and  see  whether  their  respective 
quantities  agree  with  the  rule. 

354.  A  circular  grass-plot  has  a  diameter  oi 
300  feet,  and  a  walk  of  3  yards  wide  round  it ; 
find  the  area  of  the  grass-plot,  and  also  the  area 
of  the  walk. 

355.  Can  you  find  the  area  of  a  sector  whose 
radial  boundaries  are  each  20 £  yards,  and  whose 
arc  contains  35°  \ 

350.  The  largest  pyramid  in  the  world 
stands  upon  a  square  base,  whose  side  is  700 
feet  long.  The  pyramid  has  four  equilateral 
triangles  for  its  surfaces.  Calculate  what  num- 
ber of  square  feet,  square  yards,  and  acres,  the 
base  of  such  pyramid  stands  upon,  and  the  num- 
ber of  square  feet  on  each  of  its  triangular  sur- 
faces; calculate  also  its  perpendicular  height, 
and  prove  its  correctness  by  geometry ;  give  ip 


80  INVENT10SAL    GEOMETRY 

card  a  model  of  the  pyramid ;  say  what  solid  it 
is  a  part  of,  and  give  a  sketch  of  the  model. 

357.  There  is  a  rhomboid  of  such  a  form 
that  its  area  may  be  found  by  means  of  one  of 
its  sides,  and  one  of  its  diagonals.     Give  a  plan 
of  it. 

358.  Can  you  convert  a  square  whose  side  is 
1  into  a  rhombus  whose  long  diagonal  is  twice 
as  much  as  the  short  one;  and  can  you  find, 
both  by  geometry  and  by  calculation,  the  length 
of  the  side  of  that  rhombus  ? 

359.  Can  you  convert  an  equilateral  triangle 
into  an  irregular  pentagon  ? 

360.  Point  out  upon  a  tetrahedron  two  lines 
that  are  in  the  same  plane,  and  two  that  are  not 
in  the  same  plane. 

361.  Make  of  card  a  truncated  octahedron, 
and  give  a  plan  of  it,  and  a  sketch  of  the  figure 

362.  Show  how  many  cubes  may  be  made 
to  touch  at  one  point. 

363.  Show  by  a  figure  how  many  cubes  may 
be  made  to  touch  one  cube. 


81 

You  have  calculated  the  perpendicular 
height  of  an  equilateral  triangle  whose  side  ia 
1 ;  can  you  say  how  far  up  that  perpendicular 
it  is  from  the  base  to  the  centre  of  the  tri 
angle  ? 

A  solid  formed  by  revolving  a  rectangle 
about  one  of  its  sides  takes  the  name  of  cylin- 
der,1 and  it  may  be  called  a  circular  prism. 

365.  Can  you  find  the  surface  of  the  cylin- 
der whose  length  is  1,  and  whose  diameter  is  1 J 

A  sphere  may  be  formed  by  revolving  a 
semicircle  about  the  diameter  as  an  axis. 

The  surface  of  a  sphere  whose  diameter  ia 
1,  is  equal  to  the  surface  of  a  cylinder  whose 
diameter  is  1  and  height  1.  Give  a  figure  in 
illustration  of  what  is  meant. 

366.  Find  the  surface  of  a  sphere  whose  di- 
ameter is  1,  and  also  the  surface  of  a  sphere 
whose  diameter  is  2.     Compare  the   two  sur- 
taces  together,  and  say  whether  the  ratio  the 
Less  has  to   the  greater  accords  with  the  law, 

1  When  a  cylinder  is  long  it  takes  the  name  of  rod,  as  a 
rod  of  iron. 


£2  INTENTIONAL    GEOMETRY. 

"  The  areas  of  similar  figures  are  to  each  othei 
as  the  squares  of  their  homologous  sides." 

367.  Can  you  erect  an  hexagonal  pyramid 
whose  slant  sides  shall  be  equilateral  triangles  I 

368.  Make  a  box  of  strong  pasteboard,  and 
let  the  length  be  five  inches,  breadth  four,  and 
depth  three,  and  let  it  have  a  lid  that  shall  not 
only  cover  the  box,  but  have  edges  clasping  it 
when  shut,  and  hanging  over  the  top  of  the 
box  three-eighths  of  an  inch. 

369.  Can  you  plant  19  trees  in  9  rows  of  6 
in  a  row  ? 

370.  Can  you  convert  a  scalene  triangle  into 
a  symmetrical  trapezium  J 

371.  Place  a  hexagon  inside  an  equilateral 
triangle,  so  that  three  of  its  sides  may  touch  it, 
and  show  the  ratio  the  hexagon  bears  to  the 
tringle. 

372.  A.  philosopher  had  a  window  a  yard 
square,  and  it  let  in  too  much  light ;  he  blocked 
up  one  half  of  it,  and  still  had  a  square  window 
a  yard  high  and  a  yard  wide.    Say  how  he  did  it 


GEOMETRY.  83 

373.  Can  you  divide  an  equilateral  triangle 
into  two  equal  parts  by  a  line  drawn  parallel  to 
one  of  its  sides  \ 

374.  Given  the  chord  of  an  arc  50,  and  the 
sine  of  the  arc  40 ;  required  the  versed  sine  by 
calculation,  and  point  out  on  the  figure  that  it 
is  equal  to  radius  minus  co-sine. 

375.  Can  you  divide  a  common  triangle  into 
two  equal  parts  by  a  line  parallel  to  one  of  its 
sides  ? 

376.  Can  you  divide  a  triangle  into  two 
equal  parts  by  a  line  from  any  point  in  any  one 
of  its  sides  ? 

377.  Show  how  many  solid  feet  there  are  in 
a  solid  yard. 

378.  Make  an  oblique  square  prism  with  two 
rectangular  sides  and  two  rhomboidal  sides. 

379.  Make  an  oblique  square  prism  with  all 
its  sides  equally  rhomboidal. 

380.  Can  you  place  an  equilateral  triangle 
in  a  square  so  that  one  angular  point  of  the 
equilateral  triangle  may  coincide  with  one  an 


84  INVEXTWXAA.   GEOMETRY 

gular  point  of  the  square,  and  the  other  two 
angular  points  of  the  triangle  may  touch,  at 
equal  distances  from  the  angle  of  the  square, 
two  of  the  sides  of  the  square  f 

381.  Can  you  divide  a  line  into  5  equal 
parts  ? 

382.  Can  you  divide  a  line  into  3£  parts  ? 

383.  Can  you  divide  a  line  as  any  other  line 
is  divided  ? 

384.  Can  you  answer  by  geometry  the  ques 
tion,  When  three  yards  of  cloth  cost  12*.,  what 
will  five  yards  cost  ? 

The  rule  by  which  areas  are  found,  when  the 
dimensions  are  given  in  feet  and  inches,  takes 
the  name  of  duodecimals ;  such  areas  being 
always  expressed  in  feet,  twelfths  of  feet,  or 
parts,  twelfths  of  parts,  or  square  inches.  (See 
Young's  "  Mensuration.") 

Duodecimals  are  used  chiefly  by  artisans  for 
the  purpose  of  determining  the  quantity  oi 
work  they  have  done,  or  the  quantity  of  ma 
terials  they  have  used. 


INVENTIONAL   GEOMETRY.  86 

•    385.  Give  a  plan  of  a  duodecimal  part,  that 
is,  of  a  twelfth  of  a  foot. 

386.  Give  a  plan  of  a  duodecimal  inch,  that 
g,  of  one-twelfth  of  a  part,  and  show  that  its 

size  is  the  same  as  the  square  inch,  although 
the  forms  may  differ. 

387.  Give  a  plan  that  shall  show  the  area  of 
a  square  whose  side  is  1  ft.  1  in.,  and  prove  by 
duodecimals. 

388.  Give  by  a  scale  of  an  inch  to  a  foot 
the  plan  of  a  board,  3  ft.  4  in.  long,  and  2 
ft.  2  in.  wide,  and  prove  by  duodecimals  the 
area. 

389.  Ascertain    by    geometry    how  many 
inches  there  are  in  the  diagonal  of  a  square 
foot,  and  how  many  in  the  diagonal  of  a  cubic 
foot,  and  prove  by  calculation. 

390.  Can  you  make  an  octagon  which  shall 
have  its  alternate  sides  one-half  of  the  others, 
And  that  shall  still  be  symmetrical  ? 

391.  Can  you  place  in  a  pentagon  a  rhom- 
bus that  shall   touch   with  its  angular   points 


86  ISVENTIONAL   GEOMETRY. 

three  of  the  sides  of  the  pentagon  and  one  oi 
its  angles  ? 

392.  Let  there  be  two  rectangles  of  differ- 
ent magnitudes,  but  similar  in  form  ;  it  is  re- 
quired to  determine  the  size  of  another  similar 
one  that  shall  equal  their  sum. 

393.  Given  two  triangles  dissimilar  and  un- 
equal, can  you  make  a  triangle  equal  to  their 
sum  j 

394.  Can  you  make  a  triangle  equal  to  the 
difference  of  two  triangles ! 

395.  Can  you  make  a  rectangle  equal  to  the 
difference  of  two  rectangles  ? 

396.  Can  you  place  three  circles  of  equal 
radii  to  touch  each  other  ? 

397.  Place  a  regular  octagon  in  a  square,  so 
that  four  sides  of  the  octagon  may  touch  the 
four  sides  of  the  square. 

398.  There  is  one  class  of  triangles  that  will 
divide  into  two  triangles  that  are  both  equal 
and  similar ;  there  is   another   class   that  wili 
bear  dividing  into  two  triangles  that  are  similar, 


INVENIIONAL  GEOMETRY.  g« 

but  not  equal ;  and  a  third  class  that  may  be 
divided  into  two  that  are  equal,  but  not  simi- 
lar. Give  an  example  of  each. 

399.  Can  you  divide  a  trapezium  into  two 
equal  parts  by  a  line  drawn  from  a  point  in  one 
of  the  sides  ? 

400.  Can  you  place  three  circles,  whose  di- 
ameters are  3,  4,  and  5,  to  touch  one  another  \ 

401.  Make  of  strong  cardboard  a  box  open 
at  one  end,  and  large  enough  to  receive  a  pack 
of  cards,  and  make  a  lid  that  shall  slide  on  that 
end  and  go  over  it  three-quarters  of  an  inch. 

402.  Can  you  make  a  square  that  shall  con 
naiu  three-quarters  of  another  square  ? 

403.  Can  you  place  a  square  in  an  equilat- 
eral triangle  \ 

404.  Can  you  place  a  square  in  an  isosceles 
triangle  ? 

405.  Can  you  place  a  square  in  a  quadrant  f 

406.  Can  you  place  a  square  in  a  semicircle  1 

407.  Can  you  place  a  square  in  any  triangle  ? 


Rg  INTENTIONAL   GEOMETRY. 

408.  Can  you  place  a  square  in  a  pentagon  I 

409.  Determine  the  form  of  that  rectangle 
which  will  bear  halving  by  a  line  drawn  par- 
allel to  its  shortest  side,  without  altering  its 
form.   . 

410.  Show  that  there  is  a  polygon,  the  in- 
terior of  which  may,  by  four  lines,  be  divided 
into  nine  figures ;  one  being  a  square,  four  re 
ciprocal  rectangles,  and  the  remaining  four  re- 
ciprocal triangles. 

» 

411.  Geometricians  have  asserted  that,  when 
in  a  circle  one  chord  halves  another  chord,  the 
rectangle  contained   by   the   segments   of  the 
halving  chord  is  equal  to  the  square  of  one  hali 
of  the  chord  which  is  halved  ;  and  that,  when 
one  chord  in  a  circle  halves  another  chord  at 
right  angles,  one  half  of  the  halved  chord  is  a 
mean  proportional  between  the  segments  of  the 
halving  chord.     Determine,  as  nearly  as  you 
can,  by  a  scale,  whether  it  is  true. 

One-half  of  the  sum  of  any  two  numbers 
or  any  two  lines  is  called  the  arithmetic  mean 
to  those  numbers,  or  to  those  lines. 


INVBXTIONAL  GEOMETRY.  89 

412.  Show  by  a  figure  the  arithmetical  mean 
to  3  and  12. 

The  arithmetic  mean  has  the  same  distance 
from  the  less  extreme  that  the  greater  extreme 
has  from  it. 

The  square  root  of  the  product  of  two  num- 
bers is  called  the  geometric  mean  to  those  num- 
bers. 

413.  Show  by  a  figure  the  geometric  mean 
to  3  and  12. 

The  geometric  mean  has  the  same  ratio  to 
one  extreme  that  the  other  extreme  has  to  it, 
thus  3 :  6  : :  6  :  12.  This  is  why  it  also  takes  the 
name  of  mean  proportional. 

414.  Find  the  arithmetic  mean  and  the 
geometric  mean  to  4  and  9.  Say  which  mean 
is  the  greater. 

415.  Determine  by  geometry  and  prove  by 
calculation  the  side  of  a  square  that  shall  just 
contain  an  acre. 

416.  Extract  by  geometry  the  square  root  oi 
5,  and  prove  by  arithmetic 

The  angle  which  the  chord  of  a  segment 


90  1NVENT10NAL   &E<  VETRJ. 

makes  with  the  tangent  of  the  segment  is  called 
the  angle  of  the  segment. 

417.  Can  you  determine  the  angle  of  a  seg 
ment  of  90°  ? 

418.  Can  you  determine  which   two  lines 
drawn  from  the  extremities  of  the  chord  of  a 
segment  so  as  to  meet  together  in  the  arc  of  the 
segment  will  make  the  greatest  angle  ? 

419.  Can  you  determine  the  angle  in  a  quad- 
rantal  segment  ? 

420.  Can  you  ascertain  the  relation  existing 
betwixt  the  angle  of  a  segment  and  the  angle 
in  a  segment  ? 

421.  Can  you  give  an  instance  where  the 
angle  in  the  segment  and  the  angle  of  the  seg- 
ment are  equal  3 

A  line  that  begins  outside  a  circle,  and  on 
being  produced  enters  it,  and  traverses  it  until 
stopped  by  the  other  side  of  it,  is  called  a  secant 
to  a  circle. 

422.  Make  a  few  circles  and  fit  a  secant  to 
each. 


USVENTWXAL   GEOMETRY.  9] 

A  line  drawn  from  the  centre  of  a  circle 
through  one  extremity  of  an  arc  until  inter- 
cepted by  a  tangent  drawn  from  the  other  ex- 
tremity is  called  the  secant  of  that  arc. 

423.  Can  you  make  the  secant  of  the  arc 
of  60° \ 

Like  the  word  tangent,  secant  has  two  mean- 
ings, one  as  applied  to  a  circle,  and  the  other  as 
applied  to  an  arc. 

Can  you  on  one  line,  and  beginning  at  one 
point  in  that  line,  place  the  secants  of  all  the 
arcs  from  10  to  80° )  In  other  words — 

4:24:.  Can  you  make  a  line  of  secants  I 

425.  Make  and  measure  a  few  angles  by  the 
line  of  secants. 

426.  Say  which  you  consider  the  most  con- 
venient for  plotting  and  for  measuring  angles, 
the  line  of  chords,   of  tangents,  of  sines,  of 
versed  sines,  or  of  secants. 

427.  Calculate  the  length  of  a  link  of  a  land 
chain,  and  give  on  paper  an  exact  drawing  of 
•uch  a  link. 


92  IXVENTIONAL   UtiOMETRY. 

You  have  determined  how  far  up  th'e  perpen 
dicular  of  an  equilateral  triangle  the  centre  is. 

428.  What  ratio  have  the  two  parts  of  an 
equilateral  triangle  which  are  made  by  a  line 
drawn  through  the  centre  of  the  triangle  paral- 
lel to  the  base  ? 

429.  Suppose  the  side  of  a  hexagon  to  be  1, 
it  is  required  to  determine  the  sides  of  a  rect- 
angle that  shall  exactly  inclose  it,  and  to  find 
the  area  of  the  hexagon  and  the  area  of  the 
rectangle,  and  the  ratio  between  them. 

430.  Make  a  figure  that  shall  be  equal  to  one 
formed  by  three  squares  placed  at  an  angle,  thus 
gj;  and  say  whether  it  is  possible  to  divide  such 
tigure  into  four  equal  and  similar  parts. 

A  right-angled  triangle  made  to  revolve 
about  one  of  the  sides  containing  the  right  an- 
gle forms  a  body  called  a  cone,  which  may  be 
very  properly  called  a  circular  pyramid. 

431.  Make  in  paper  a  hollow  cone,  and  give  a 
plan  of  your  method. 

When  a  cone  is  cut  by  a  plane  at  right  an 
gleg  to  the  axis,  the  section  produced  is  a  circle. 


INVENTION AL   GEOMETRY.  93 

When  a  cone  is  cut  by  a  plane  that  makes 
the  axis  an  angle  that  is  less  than  a  right 
angle,  but  not  so  small  an  angle  as  the  angle 
which  the  side  of  the  cone  makes  with  it,  snch 
section  is  an  ellipse. 

A  section  of  a  cone  making  an  angle  with 
the  axis  equal  to  that  which  the  side  makes  is  a 
parabola. 

A  section  of  a  cone  which  makes,  with  the 
axis,  an  angle  that  is  less  than  that  which  the 
side  makes  is  an  hyperbola. 

A  section  of  a  cone  with  which  the  axis  coin- 
cides is  an  isosceles  triangle. 

432.  Cut  from  an  apple  or  a  turnip  as  accu- 
rate a  cone  as  you  can,  and  give  a  specimen  of 
each  of  the  five  conic  sections. 

433.  Give  a  sketch  of  a  builder's  trammel, 
and  make  an  ellipse  with  a  trammel ;  and  show 
that  you  can,  on  the  principle  on  which  it  acts, 
make  an  ellipse  without  one. 

The  long  diameter  of  an  ellipse  is  called  the 
axis  major,  and  the  short  one  the  axis  minor, 
»nd  the  distance  of  either  of  the  foci  of  the 


94  1AVENT10NAL 

ellipse  from  its  centre  is  called  the  eccentricity 
of  the  ellipse. 

434.  Give  a  figure  to  show  what  is  meant 
Two  lines  drawn  from  the  foci  of  an  ellipse 

to  a  point  in  the   circumference   make   equal 
angles  with  a  tangent  to  the  ellipse  at  that  point. 

435.  Can  you  at  a  particular  point  in  the 
circumference  of  an  ellipse  draw  a  tangent  to 
that  circumference  \ 

With  a  pair  of  compasses,  and  using  differ- 
ent-sized circles,  how  nearly  can  you  imitate  an 
ellipse  \  In  other  words — 

436.  How  would  you  make  an  oval  f 

437.  Can  you,  out  of  a  circular  piece  of  ma- 
hogany, and  without  any  loss,  make  the  tops  of 
two  oval  stools,  with  an  opening  to  lift  it  by,  in 
the  middle  of  each  ? 

The  solid  formed  by  revolving  on  its  minor 
axis  a  semi-ellipse  is  called  an  oblate  spheroid. 

438.  Show  how  an  oblate  spheroid  is  formed, 
and  say  what  the  oblate  spheroid  reminds  you 
of 


UEuMETRY.  96 

The  solid  formed  by  revolving  OL  ita  axis 
major  oiie-half  an  ellipse  is  called  a  prolate 
spheroid. 

439.  Show  how  a  prolate  spheroid  is  formed* 
and  say  what  it  reminds  you  of. 

440.  Supposing  a  room  to  be  built  in  the 
form  of  a  prolate  spheroid,  and  a  person  to 
speak  from  one  focus,  show  where  his  voice 
would  be  reflected. 

441.  Would  there  be  the  same  effect  pro- 
duced in  a  room  built  in  the  form  of  an  oblate 
spheroid. 

Provided  no  notice  is  taken  of  the  resist- 
ance of  the  air,  a  stone  thrown  horizontally 
from  the  top  of  a  tower,  at  a  velocity  of  48  ft. 
in  a  second,  and  subject  to  the  incessant  action 
of  the  earth,  which  from  nothing  induces  it  to 
fall  by  a  uniformly-increasing  velocity  through 
about  16  ft.  in  the  first  second,  48  ft.  in  the 
second  second,  80  ft.  in  the  third  second,  112 
ft  in  the  fourth  second,  and  so  on,  makes  in  its 
progress  a  kind  of  curve.  Kow,  the  terms  of 
the  series  16,  48,  8u,  112,  144,  etc.,  increase  in 


tf  INVENT10NAL    GEOMETRY. 

a  certain  ratio;  and,  if  16  be  called  1,  48  will 
be  3,  80  will  be  5,  112  will  be  7,  and  144 
will  be  9,  etc.  These  distances  may  then  be 
expressed  as  falling  distances,  thus,  1,  3,  5,  7,  9, 
etc.  And,  keeping  in  mind  that  the  horizontal 
velocity  remains  uniform,  that  is  48  ft.,  i.  e., 
3  x  16  ft.  in  a  second,  we  have  two  kinds  of 
dimensions  at  right  angles  to  each  other,  from 
which  to  make  the  curve.  This  curve  is  called 
a  parabola. 

442.  Can  you  construct  a  parabola  ? 

When  these  distances,  instead  of  being  writ- 
ten down  as  the  separate  result  of  each  second's 
action,  are  successively  added  to  show  the  com- 
bined results,  we  have  for  — 


Distance*. 

1  second  ...........................       1     =     1* 

2  seconds,  1+8=  ..........  .  ........       4     =     2* 

3  seconds,  1-h  3  +  5=  ...............         0     =     8* 

4  seconds,  1-1-8  +  6  +  7=  .............     16     =     4* 

5  seconds,  1  -1-3  +  5+7+9=  ..........     25     -     5» 

From  this  it  will  be  seen  that  the  distance 
fallen  is  as  the  square  of  the  time,  i.  e.,  in  t 


INVEtillONAL    GEOMETRY.  97 

•ecotids  the  distance  fallen  will  be  6*  x  16  ft.  = 
36  X  16  ft.  ==  576  ft. 

443.  Required  the  distance  a  stone  falls  in 
half  a  second. 

444.  Required  the  distance  a*  stone  falls  in 
2i  seconds. 

445.  Can  you  show  that  there  are  two  kinds 
of  quadrilaterals  in  which  the  diagonals  must 
be  equal,  two  kinds  where  they  may  be  equal, 
and  two  kinds  where  they  cannot  be  equal  ? 

446.  Can  you  make  in  card  a  tetrahedron 
whose  four  surfaces  shall  be  unlike  in  form  f 


1H£ 


A    HISTORY    OF    ENGLISH 
LITERATURE 

By   REUBEN    POST    HALLECK,    M.A.    (Yale), 
Louisville  Male  High  School.      Price,  $1.2$ 


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GRAMMAR  AND  RHETORIC 
FOR  SECONDARY  SCHOOLS 

By  MARIETTA  KNIGHT,  English  Department,  South 
High  School,  Worcester,  Mass.      Price,  25  cents 


THIS  primer  is  the  outcome  of  the  need  felt  by  many 
teachers  in  secondary  schools  for  a  concise  and  com- 
pact summary  of  the  essentials  of  grammar  and  rhetoric. 
^j  It  is  designed  as  a  guide  in  review  study  of  the  ordinary 
text-books  of  grammar  and  rhetoric,  or  as  an  aid  to  teachers 
who  dispense  with  such  text-books;  in  either  case  it  is 
assumed  that  abundant  drill  work  has  been  provided  by  the 
teacher  in  connection  with  each  subject  treated. 
^[  The  work  will  also  be  found  to  harmonize  well  with 
the  recommendations  of  the  College  Entrance  Examination 
Board,  which  require  that  students  should  be  familiar  with 
the  fundamental  principles  of  grammar  and  rhetoric. 
^[  The  book  is  divided  as  follows  : 

^f  First. — Rules,  definitions,  and  principles  of  English 
grammar.  Here  there  are  treated  with  great  clearness  not 
only  the  various  parts  of  speech,  but  also  sentences,  clauses, 
phrases,  capitals,  and  punctuation. 

^j  Second. — Rules,  definitions,  and  principles  of  rhetoric. 
This  part  of  the  book  takes  up  the  forms  of  composition, 
narration,  description,  exposition,  and  argument,  letter- 
writing,  the  paragraph,  the  sentence,  choice  and  use  of 
words,  figures  of  speech,  and  poetry,  the  various  kinds  of 
meters,  etc.  At  the  close  there  is  a  brief  collection  of 
"Don'ts,"  both  rhetorical  and  grammatical,  many  "  Helps 
in  Writing  a  Theme/'  and  a  very  useful  index. 


AMERICAN    BOOK   COMPANY 

(S.  82) 


A  PUNCTUATION  PRIMER 

By    FRANCES    M.    PERRY,    Associate   Professor   of 
Rhetoric    and    Composition,    Wellesley    College. 

$0-30 


THE  Punctuation  Primer  is  a  manual  of  first  principles 
or  essentials  simply  and  systematically  presented;    it 
is  not  an  elaborate  treatise  on  punctuation.     It  offers 
a  few  fundamental  principles  that  are  flexible  and  com- 
prehensive, and  easily  understood  and  remembered.     The 
meaning  of  the  text  to  be  punctuated  and  the  grammatical 
structure  of  the  sentence  are  made  the  bases  for  general- 
ization and  division. 

^[  The  discussion  is  taken  up  under  two  main  divisions: 
The  terminal  punctuation  of  sentences,  and  the  punctuation 
of  elements  within  sentences.  Under  punctuation  of 
elements  within  sentences,  the  punctuation  of  principal 
elements,  of  dependent  elements,  of  coordinate  elements, 
of  independent  elements,  and  of  implied  elements  are 
considered  in  the  order  given. 

^[  In  addition,  several  important  related  topics  are  treated, 
such  as  paragraphing,  quotations,  capitalization,  compound 
words,  word  divisions,  the  uses  of  the  apostrophe,  the 
preparation  and  the  correction  of  manuscript,  conventional 
forms  for  letters,  the  use  of  authorities  in  writing  themes, 
the  correction  of  themes,  and  the  making  of  bibliographies. 
^[  Throughout  the  carefully  selected  examples  make  clear 
the  meaning  of  the  text,  while  the  exercises  provided  at 
each  stage  of  the  work  afford  the  student  practice  in  the 
correct  application  of  the  principles. 

^j  Though  written  primarily  to  meet  the  needs  of  college 
freshmen,  the  primer  is  an  excellent  manual  for  high  schools. 


AMERICAN  BOOK  COM  PANT 

(S.84) 


NINETEENTH     CENTURY 
ENGLISH     PROSE 

Critical    Essays 

Edited  with  Introductions  and  Notes  by  THOMAS  H. 
DICKINSON,  Ph.D.,  and  FREDERICK  W,  ROE, 
A.M.,  Assistant  Professors  of  English,  University  of 
Wisconsin.  Price,  $1.00. 


THIS  book  for  college  classes  presents  a  series  often 
selected  essays,  which  are  intended  to  trace  the 
development  of  English  criticism  in  the  nineteenth 
century.  The  essays  cover  a  definite  period,  and  exhibit 
the  individuality  of  each  author's  method  of  criticism.  In 
each  case  they  are  those  most  typical  of  the  author's  crit- 
ical principles,  and  at  the  same  time  representative  of  the 
critical  tendencies  of  his  age.  The  subject-matter  provides 
interesting  material  for  intensive  study  and  class  room  dis- 
cussion, and  each  essay  is  an  example  of  excellent,  though 
varying,  style. 

5[  They  represent  not  only  the  authors  who  write,  but 
the  authors  who  are  treated.  The  essays  provide  the 
best  things  that  have  been  said  by  England's  critics  on  Swift, 
on  Scott,  on  Macaulay,  and  on  Emerson. 

^[  The  introductions  and  notes  provide  the  necessary  bio- 
graphical matter,  suggestive  points  for  the  use  of  the  teacher 
in  stimulating  discussion  of  the  form  or  content  of  the  essays, 
and  such  aids  as  will  eliminate  those  matters  of  detail  that 
might  prove  stumbling  blocks  to  the  student.  Though  the 
essays  are  in  chronological  order,  they  may  be  treated  at 
random  according  to  the  purposes  of  the  teacher. 


AMERICAN     BOOK     COMPANY 

(S.8o) 


INTRODUCTORY  COURSE 
IN   EXPOSITION 

By    FRANCES    M.    PERRY,    Associate    Professor   of 
Rhetoric    and    Composition,    Wellesley   College. 


i.oo 


EXPOSITION  is  generally  admitted  to  be  the  most 
commonly  used  form  of  discourse,  and  its  successful 
practice  develops  keen  observation,  deliberation, 
sound  critical  judgment,  and  clear  and  concise  expression. 
Unfortunately,  however,  expository  courses  often  fail  to 
justify  the  prevailing  estimate  of  the  value  of  exposition, 
because  the  subject  has  been  presented  in  an  unsystem- 
atized  manner  without  variety  or  movement. 
^[  The  aim  of  this  book  is  to  provide  a  systematized 
course  in  the  theory  and  practice  of  expository  writing. 
The  student  will  acquire  from  its  study  a  clear  under- 
standing of  exposition  —  its  nature  ;  its  two  processes, 
definition  and  analysis ;  its  three  functionSj  impersonal 
presentation  or  transcript,  interpretation,  and  interpretative 
presentation  ;  and  the  special  application  of  exposition  in 
literary  criticism.  He  will  also  gain,  through  the  practice 
required  by  the  course,  facility  in  writing  in  a  clear  and 
attractive  way  the  various  types  of  exposition.  The 
volume  includes  an  interesting  section  on  literary  criticism. 
^|  The  method  used  is  direct  exposition,  amply  reinforced 
by  examples  and  exercises.  The  illustrative  matter  is 
taken  from  many  and  varied  sources,  but  much  of  it  is 
necessarily  modern.  The  book  meets  the  needs  of 
students  in  the  final  years  of  secondary  schools,  or  the 
first  years  of  college. 


AMERICAN  BOOK  COM  PA  NY 

(S-93) 


INTRODUCTORY  COURSE 
IN  ARGUMENTATION 

By  FRANCES    M.    PERRY,  Instructor  in  English  in 
Wellesley  College.      Price,   $1.00 


AN  INTRODUCTORY  COURSE   IN  ARGU- 
MENTATION is  intended  for  those  who  have  not 
previously  studied  the  subject,  but  while  it  makes  a 
firm  foundation  for  students  who  may  wish  to  continue  it, 
the  volume  is  complete  in  itself.      It  is  adapted  for  use  in 
the  first  years  of  college  or  in  the  upper  classes  of  second- 
ary schools. 

^[  The  subject  has  been  simplified  as  much  as  has  been  pos- 
sible without  lessening  its  educative  value,  yet  no  difficul- 
ties have  been  slighted.  The  beginner  is  set  to  work  to 
exercise  his  reasoning  power  on  familiar  material  and  with- 
out the  added  difficulty  of  research.  Persuasion  has  not  been 
considered  until  conviction  is  fully  understood.  The  two 
methods  in  use  in  teaching  argumentation — the  brief-draw- 
ing method  and  the  syllogistic  method — have  been  com- 
bined, so  that  the  one  will  help  the  student  to  grasp  the  other.. 
^|  The  volume  is  planned  and  proportioned  with  the  ex- 
pectation that  it  will  be  closely  followed  as  a  text-book 
rather  than  used  to  supplement  an  independent  method  of 
presentation.  To  that  end  each  successive  step  is  given  ex- 
plicit exposition  and  full  illustration,  and  carefully  graded 
exercises  are  provided  to  test  the  student's  understanding 
of  an  idea,  and  fix  it  in  his  memory. 
^[  The  course  is  presented  in  three  divisions  ;  the  first  re- 
lating to  finding  and  formulating  the  proposition  for  argu- 
ment, the  second  to  proving  the  proposition,  and  the  last, 
to  finding  the  material  to  prove  the  proposition — research. 


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NEW  ROLFE  SHAKESPEARE 

Edited  by  WILLIAM  J.   ROLFE,  Litt.D. 
40  volumes,  each,  $0.56 


THE  popularity  of  Rolfe's  Shakespeare  has  been  ex- 
traordinary. Since  its  first  publication  in  1870-83 
it  has  been  used  more  widely,  both  in  schools  and 
colleges,  and  by  the  general  reading  public,  than  any  simi- 
lar edition  ever  issued.  It  is  to-day  the  standard  annotated 
edition  of  Shakespeare  for  educational  purposes. 
^[  As  teacher  and  lecturer  Dr.  Rolfe  has  been  constantly  in 
touch  with  the  recent  notable  advances  made  in  Shakespear- 
ian investigation  and  criticism  ;  and  this  revised  edition  he 
has  carefully  adjusted  to  present  conditions. 
^j  The  introductions  and  appendices  have  been  entirely  re- 
written, and  now  contain  the  history  of  the  plays  and  poems; 
an  account  of  the  sources  of  the  plots,  with  copious  extracts 
from  the  chronicles  and  novels  from  which  the  poet  drew 
his  material  ;  and  general  comments  by  the  editor,  with 
selections  from  the  best  English  and  foreign  criticism. 
^j  The  notes  are  very  full,  and  include  all  the  historical, 
critical,  and  illustrative  material  needed  by  the  teacher,  as 
well  as  by  the  student,  and  general  reader.  Special  feat- 
ures in  the  notes  are  the  extent  to  which  Shakespeare  is 
made  to  explain  himself  by  parallel  passages  from  his  works; 
the  frequent  Bible  illustrations;  the  full  explanations  of  allu- 
sions to  the  manners  and  customs  of  the  period;  and  descrip- 
tions of  the  localities  connected  with  the  poet' s  life  and  works0 
^[  New  notes  have  also  been  substituted  for  those  referring 
to  other  volumes  of  the  edition,  so  that  each  volume  is  now 
absolutely  complete  in  itself.  The  form  of  the  books  has 
been  modified,  the  page  being  made  smaller  to  adjust  them 
to  pocket  use. 


AMERICAN    BOOK    COMPANY 

(S.  97) 


GATEWAY 
SERIES  OF  ENGLISH  TEXTS 

General     Editor,     HENRY    VAN    DYKE,     Princeton 

University 


INCLUDES  the  English  texts  required  for  entrance  to 
college,  in  a  form  which  makes  them  clear,  interesting 
and  helpful  in  beginning  the  study  of  literature. 

Addison's  Sir  Roger  de  Coverley  Papers  (Winchester)  .     .      .     $0.40 

Burke's  Speech  on  Conciliation  (Mac  Donald) .35 

Byron,  Wordsworth,  Shelley,  Keats  and  Browning — Selections 

(Copeland  &  Rideout) .40 

Carlyle's  Essay  on  Burns  (Mims) .35 

Coleridge's  Rime  of  the  Ancient  Mariner  (Woodberry).      .      .          .30 

Emerson's  Essays — Selections  (Van  Dyke) .35 

Franklin's  Autobiography  (Smyth) .40 

Gaskell's  Cranford  (Rhodes) 40 

George  Eliot's  Silas  Marner  (Cross) .40 

Goldsmith's  Vicar  of  Wakefield,  and  The  Deserted   Village 

(Tufts) 45 

Irving's  Sketch  Book — Selections  (Sampson)       .....          .45 

Lamb's  Essays  of  Elia  (Sampson) .40 

Macaulay's  Essay  on  Addison  (McClumpha) .35 

Essay  on  Milton  (Gulick) .35 

Life  of  Johnson  (Clark) .35 

Addison  and  Johnson.       One  Volume.       (McClumpha- 

Clark) 45 

Milton's  Minor  Poems  (Jordan) .35 

Scott's  Ivanhoe  (Stoddard) 50 

Lady  of  the  Lake  (Alden) .40 

Shakespeare's  As  You  Like  It  (Demmon) .35 

Julius 'Caesar  (Mabie) 35 

Macbeth  (Parrott) .40 

Merchant  of  Venice  (Schelling) .35 

Tennyson's  Idylls  of  the  King — Selections  (Van  Dyke)      .      .          .35 

Princess  (Bates) 40 

Washington's  Farewell  Address,  and  Webster's  First  Bunker 
Hill  Oration  (Kent) 


AMERICAN     BOOK     COMPANY 

(S.99) 


AMERICAN     LITERATURE 


AMERICAN  POEMS #0.90 

With  notes  and  biographies.  By  AUGUSTUS 
WHITE  LONG,  Preceptor  in  English,  Princeton 
University,  Joint  Editor  of  Poems  from  Chaucer  to 
Kipling 

THIS  book  is  intended  to  serve  as  an  introduction  to  the 
systematic  study  of  American  poetry,  and,  therefore, 
does  not  pretend  to  exhaustiveness.  All  the  poets 
from  1 776  to  1900  who  are  worthy  of  recognition  are  here 
treated  simply,  yet  suggestively,  and  in  such  a  manner  as  to 
illustrate  the  growth  and  spirit  of  American  life,  as  ex- 
pressed in  its  verse.  Each  writer  is  represented  by  an 
appropriate  number  of  poems,  which  are  preceded  by  brief 
biographical  sketches,  designed  to  entertain  and  awaken 
interest.  The  explanatory  notes  and  the  brief  critical 
comments  give  much  useful  and  interesting  information. 


MANUAL  OF  AMERICAN  LITERATURE,  $0.60 

By  JAMES  B.  SMILEY,  A.M.,  Assistant  Principal 
of  Lincoln  High  School,  Cleveland,  Ohio 

THE  aim  of  this  little  manual  is  simply  to  open  the  way 
to  a  study  of  the  masterpieces  of  American  literature. 
The  treatment  is  biographical  rather  than  critical,  as 
the  intention  is  to  interest  beginners  in  the  lives  of  the  great 
writers.      Although  the  greatest  space  has  been  devoted  to 
the  most  celebrated  writers,  attention  is  also  directed  to 
authors  prominent  in  the  early  history  of  our  country,  and 
to  a  few  writers  whose  books   are  enjoying  the  popularity 
of  the  moment.      Suggestions  for  reading  appear  at  the  end 
of  the  chapters. 


AMERICAN      BOOK      COMPANY 


THE   MASTERY  OF  BOOKS 

By  HARRY  LYMAN  KOOPMAN,  A.M.,  Librarian 
of  Brown  University.      Price,  90  cents 


IN   this   book   Mr.   Koopman,    whose  experience  and 
reputation  as  a  librarian  give  him  unusual  qualifications 
as  an  adviser,  presents  to  the  student  at  the  outset  the 
advantages  of  reading,   and  the   great  field    of  literature 
open  to  the  reader's  choice.     He  takes  counsel  with  the 
student  as  to  his  purpose,  capacities,  and  opportunities  in 
reading,  and  aims  to  assist  him  in  following  such  methods 
and  in  turning  to  such  classes  of  books  as  will  further  the 
attainment  of  his  object. 

^j  Pains  are  taken  to  provide  the  young  student  from  the 
beginning  with  a  knowledge,  often  lacking  in  older  readers, 
of  the  simplest  literary  tools — reference  books  and  cata- 
logues. An  entire  chapter  is  given  to  the  discussion  of 
the  nature  and  value  of  that  form  of  printed  matter  which 
forms  the  chief  reading  of  the  modern  world — periodical 
literature.  Methods  of  note- taking  and  of  mnemonics 
are  fully  described ;  and  a  highly  suggestive  and  valuable 
chapter  is  devoted  to  larguage  study. 
^[  One  of  the  most  valuable  chapters  in  the  volume  to 
most  readers  is  that  concerning  courses  of  reading.  In 
accordance  with  the  author's  new  plan  for  the  guidance 
of  readers,  a  classified  list  of  about  fifteen  hundred  books 
is  given,  comprising  the  most  valuable  works  in  reference 
books,  periodicals,  philosophy,  religion,  mythology  and 
folk-lore,  biography,  history,  travels,  sociology,  natural 
sciences,  art,  poetry,  fiction,  Greek,  Latin,  and  modern 
literatures.  The  latest  and  best  editions  are  specified,  and 
the  relative  value  of  the  several  works  mentioned  is  indi- 
cated in  notes. 


AMERICAN    BOOK   COMPANY 

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HISTORY  OF  ENGLISH  AND 
AMERICAN  LITERATURE 

By  CHARLES  F.  JOHNSON,  L.H.D.,  Professor  of 
English  Literature,  Trinity  College,  Hartford.     Price, 

$1.25 

A  TEXT-BOOK  for  a  year's  course  in  schools  and 
colleges,  in  which  English  literary  history  is  regarded 
as  composed  of  periods,  each  marked  by  a  definite 
tone  of  thought  and  manner  of  expression.  The  treatment 
follows  the  divisions  logically  and  systematically,  without 
any  of  the  perplexing  cross  divisions  so  frequently  made. 
It  is  based  on  the  historic  method  of  study,  and  refers 
briefly  to  events  in  each  period  bearing  on  social  devel- 
opment, to  changes  in  religious  and  political  theory, 
and  even  to  advances  in  the  industrial  arts.  In  addi- 
tion, the  book  contains  critiques,  general  surveys,  sum- 
maries, biographical  sketches,  bibliographies,  and  suggestive 
questions.  The  examples  have  been  chosen  from 
poems  which  are  generally  familiar,  and  of  an  illustrative 
character. 


JOHNSON'S    FORMS    OF    ENGLISH    POETRY 

$1.00 

THIS  book  contains  nothing  more  than  every  young  person  should 
know    about    the   construction  of  English  verse,  and    its   main 
divisions,  both  by  forms  and  by  subject-matter.     The  historical 
development  of  the  main  divisions  is  sketched,  and  briefly  illustrated  by 
representative  examples ;  but  the  true  character  of  poetry  as  an  art  and 
as  a  social  force   has  always  been    in   the  writer's  mind.      Only  the 
elements  of  prosody  are  given.      The  aim  has  been   not   to  make  the 
study  too  technical,  but  to  interest  the  student  in  poetry,  and  to  aid  him 
in  acquiring  a  well-rooted  taste  for  good  literature. 


AMERICAN    BOOK    COMPANY 

'S.  101) 


OUTLINES  OF  GENERAL 
HISTORY 

By  FRANK  MOORE  COLBY,  M.  A.,  recently  Pro- 
fessor of  Economics,  New  York  University 

$1-50 


THIS  volume  provides  at  once  a  general  foundation 
for  historical  knowledge  and  a  stimulus  for  further 
reading.  It  gives  each  period  and  subject  its 
proper  historical  perspective,  and  provides  a  narrative 
which  is  clear,  connected,  and  attractive.  From  first  to 
last  only  information  that  is  really  useful  has  been  included. 
^|  The  history  is  intended  to  be  suggestive  and  not 
exhaustive.  Although  the  field  covered  is  as  wide  as 
possible,  the  limitations  of  space  have  obliged  the  writer  to 
restrict  the  scope  at  some  points;  this  he  has  done  in  the 
belief  that  it  is  preferable  to  giving  a  mere  catalogue 
of  events.  The  chief  object  of  attention  in  the  chapters 
on  mediaeval  and  modern  history  is  the  European  nations, 
and  in  treating  them  an  effort  has  been  made  to  trace  their 
development  as  far  as  possible  in  a  connected  narrative, 
indicating  the  causal  relations  of  events.  Special  emphasis 
is  given  to  the  great  events  of  recent  times. 
^j  The  book  is  plentifully  supplied  with  useful  pedagogical 
features.  The  narrative  follows  the  topical  manner  of 
treatment,  and  is  not  overcrowded  with  names  and  dates. 
The  various  historical  phases  and  periods  are  clearly  shown 
by  a  series  of  striking  progressive  maps,  many  of  which 
are  printed  in  colors.  The  illustrations  are  numerous  and 
finely  executed.  Each  chapter  closes  with  a  summary  and 
synopsis  for  review,  covering  all  important  matters. 


AMERICAN    BOOK    COMPANY 

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ESSENTIALS  IN  HISTORY 


ESSENTIALS  IN  ANCIENT  HISTORY  .     11.50 

From  the  earliest  records  to  Charlemagne.  By 
ARTHUR  MAYER  WOLFSON,  Ph.D.,  First 
Assistant  in  History,  DeWitt  Clinton  High  School, 
New  York 

ESSENTIALS   IN    MEDIAEVAL  AND  MODERN 
HISTORY £1.50 

From  Charlemagne  to  the  present  day.  By  SAMUEL 
BANNISTER  HARDING,  Ph.D.,  Professor  of 
European  History,  Indiana  University 

ESSENTIALS  IN  ENGLISH  HISTORY    .     $1.50 

From  the  earliest  records  to  the  present  day.  By 
ALBERT  PERRY  WALKER,  A.M.,  Master  in 
History,  English  High  School,  Boston 

ESSENTIALS  IN  AMERICAN  HISTORY .     $  i .  50 

From  the  discovery  to  the  present  day.  By  ALBERT 
BUSHNELL  HART,  LL.D.,  Professor  of  History, 
Harvard  University 

THESE  volumes  correspond  to  the  four  subdivisions 
required  by  the  College  Entrance  Examination 
Board,  and  by  the  New  York  State  Education  De- 
partment. Each  volume  is  designed  for  one  year's  work. 
Each  of  the  writers  is  a  trained  historical  scholar,  familiar 
with  the  conditions  and  needs  of  secondary  schools. 
^j  The  effort  has  been  to  deal  only  with  the  things  which 
are  typical  and  characteristic;  to  avoid  names  and  details 
which  have  small  significance,  in  order  to  deal  more  justly 
with  the  forces  which  have  really  directed  and  governed 
mankind.  Especial  attention  is  paid  to  social  history. 
^j  The  books  are  readable  and  teachable,  and  furnish  brief 
but  useful  sets  of  bibliographies  and  suggestive  questions. 
No  pains  have  been  spared  by  maps  and  pictures  to  furnish 
a  significant  and  thorough  body  of  illustration. 


AMERICAN     BOOK     COMPANY 

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GREEK    AND    ROMAN 
HISTORIES 

By  WILLIAM  C.  MOREY,  Professor  of  History  and 

Political  Science,  University  of  Rochester 

Each,  $1.00 


MOREY' S  OUTLINES  OF  GREEK  HISTORY, 
which  is  introduced  by  a  brief  sketch  of  the  pro- 
gress of  civilization  before  the  time  of  the  Greeks 
among  the  Oriental  peoples,  pays  greater  attention  to  the 
civilization  of  ancient  Greece  than  to  its  political  history. 
The  author  has  endeavored  to  illustrate  by  facts  the  most 
important  and  distinguishing  traits  of  the  Grecian  char- 
acter; to  explain  why  the  Greeks  failed  to  develop  a 
national  state  system,  although  successful  to  a  consider- 
able extent  in  developing  free  institutions  and  an  organized 
city  state;  and  to  show  the  great  advance  made  by  the 
Greeks  upon  the  previous  culture  of  the  Orient. 
<jj  MOREY'S  OUTLINES  OF  ROMAN  HISTORY 
gives  the  history  of  Rome  to  the  revival  of  the  empire  by 
Charlemagne.  Only  those  facts  and  events  which  illus- 
trate the  real  character  of  the  Roman  people,  which  show 
the  progressive  development  of  Rome  as  a  world  power, 
and  which  explain  the  influence  that  Rome  has  exercised 
upon  modern  civilization,  have  been  emphasized.  The 
genius  of  the  Romans  for  organization,  which  gives  them 
their  distinctive  place  in  history,  is  kept  prominently  in 
mind,  and  the  kingdom,  the  republic,  and  the  empire  are 
seen  to  be  but  successive  stages  in  the  growth  of  a  policy 
to  bring  together  and  organize  the  various  elements  of  the 
ancient  world. 


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